I did not notice where any of the sources I read through provides a rule for when -σύνη {-sunê} is supposed to be used for creating a qualitative abstract noun out of an adjective, and when something else (like {-tês} or {-ia}) should be used instead. This opens the door to {-sunê} being used as a qualitative suffix for some specific purpose compared to other endings, and so opens the door again to the term being in fact directly related to {sun}. (Unless, which is entirely possible, I missed an explanation somewhere for why {-sunê} instead of, say, {-ia} ought to be used for such a conjugation of {dikaion} apart from specific meanings of the different suffixes.)
(The fact that the opposite is {adikia} instead of {adikaiosunê}, is something I have noticed myself in the past, and will volunteer for evidence pro or con in the dispute. 
It may be evidence that there is no meaningful difference intended between using {-sunê} and {-ia} as suffixes for creating qualitative nouns. Relatedly, a term like {aphrosunê}, which in all its NT usages definitely involves not being (well-)disposed together with persons, can feature both the suffix and the negative prefix. But this leads back to a consideration of the situations in which we find the suffix used.)
Considering how {-sunê} as a suffix is found used in NT texts, apart from {dikaiosunê}:
{hagiôsunê} shows up three times in the NT (so far as I can tell), all three of which are either Pauline or a comrade of Paul’s.
In Rom 1:4, the term is a unique variant to describe Christ as “designated Son of God in power, according to [or perhaps in accord with] (a?) spirit of holiness, out of resurrection of dead-ones”. Whether the spirit is only Christ’s personally, or a reference to the Holy Spirit (personally distinct from the Son and the Father), a union of togetherness between persons is explicitly intended. Translating this as a “spirit of holy-togetherness” would not be amiss, even if that is a title for the Holy Spirit (proceeding from the Father and the Son, first and foremost as the gift of God Most High to each other!–the first gift God can give being the gift of God Himself. But in any case, trinitarian theology or otherwise, the Holy Spirit is fundamentally concerned with the personal cooperation of God with persons–a concept clearly also connected to the HS’s common connection to punishment of those persons who sin against other persons.)
In 2 Cor 1:12, the whole point of St. Paul’s boasting is this, that he behaves in the whole world in the holiness and sincerity of God, and more hyper-abundantly so toward the Corinthians. The holiness here is explicitly about wholesome togetherness between persons.
In Heb 12:10, God disciplines those He intends to save from sin so that they may partake of His holiness. The whole context of the passage is about children of the Father coming to cooperate with Him as a family (instead of being rebellious children); the peaceable fruit of {dikaiosunê} is the expected result; and this is clearly connected to the Hebraist’s exhortation afterward that his readers should pursue peace with all (where a more common term for holiness, {hagiasmon}, is used by the way), apart from which no one shall be seeing the Lord. The holiness here is definitely and explicitly about persons cooperating in peace together, both with God and with other (if not really all??) created persons.
{agathôsunê} shows up four times in the NT (so far as I can tell), all four of which are Pauline.
In 2 Thessalonians 1:11, Paul says he is always praying for the Thessalonian congregation, that our God should be fulfilling every delight of {agathôsunê} and work of faith in power, so that the name of our Lord Jesus may be glorified in them, and them in Him, in accord with the grace of our God and Lord Jesus Christ. (This is one of those places where Jesus is called “the God” grammatically, btw. Just a reminder that I don’t spend all my time doing apologetics for Christian universalism. 
) While the immediate context might not necessarily have anything to do with the Thessalonians being in good-togetherness with other persons, the context both immediately and local has a ton to do with God acting in good togetherness with (at least!) Christians (including the faithful Thessalonians), and punishing people who are explicitly acting against good togetherness with (at least!) Christians. It is exactly “for this” that Paul says he is always praying that God should be fulfilling every delight of {agathôsunê} and work of faith in power among them. And then again, in that local preceding context, Paul is also thanking God always concerning the Thessalonians not only that their faith is flourishing but also that “the love of each one of you all into one another is increasing” (1:3). So the local (if not the immediate) context does have something to do with the Thessalonians acting in good togetherness with each other after all, for which they are being praised by Paul.
In Romans 15:14, Paul says that he is persuaded concerning his brethren in the Roman congregation that they are “bulging”(!) with {agathôsunê} and filled with entire knowledge. There isn’t any direct context of this implying “good-togetherness” between persons, except for Paul’s immediate followup that they are also able to be admonishing each other, which seems to be thanks to their {agathôsunê} (as well as their knowledge).
It may however be relevant that Paul has just said (v.8ff) that Christ has become a servant to Israel on behalf of the truth of God, to confirm the promises to the Jewish patriarchs, and for the Gentiles to glorify God for His mercy (as is it written in the OT which Paul then cites in several places). Thus despite Paul being convinced that they are bulging with {agathôsunê}, he has (vv.15-16) written very boldly to them on some points, so as to remind them again of some things regarding the gospel mission to the Gentiles. Whether this means Paul is writing to Gentile Christians on this ground of authority in Christ, or writing to Jews about the importance of the mission to the Gentiles, or (as I myself conclude from an analysis of the epistle) writing to both Jewish and Gentile Christians and so for both purposes here–either way a huge amount of the preceding epistle was about the importance of bringing Jew and Gentile together personally in goodness under Christ (as in fact the OT texts Paul just cited also directly talk about–much to the shock of many Jews, both in the times before Christ, in the time of Christ’s own earthly ministry, and afterward in Paul’s ministry.)
In Galatians 5:22, {agathôsunê} is part of the famous list of fruits of the spirit (or of the Spirit), contrasted against one of the NT lists of works of the flesh. While the list of fruits of the spirit does not in other terms necessarily have to do with togetherness between persons, the list of sins definitely has to do with evil works against persons in almost every term. Paul goes on to say in vv.25-26 that if we may be living in the Spirit (thus bearing the fruits of the Spirit), we may not become vainglorious, envying and challenging one another. Fruits of the Spirit are explicitly thus contrasted (before and after verse 22) with people mistreating each other; and if the spirit is the Holy Spirit (as even non-trinitarians sometimes agree), then the fruits cannot occur without personal cooperation between the Spirit and the created person (even though the Spirit takes the first and foremost lead in that relationship).
In Ephesians 5:9, Paul speaks on much the same topic as Gal 5:22 (and its contexts), namely on what amounts to the fruits of the Spirit (or here “the light”) contrasted to deeds of evil (vv.3-8). Not only are the deeds of evil explicitly presented as bad togetherness (if not exactly in a disputed compound term of that sort) between persons (“do not become joint partakers” with such persons, vv. 8, 10); but walking in the light as children of the light involves not merely being children of the light but personal cooperation with the light, being imitators of God as beloved children and walking in love as Christ also loves us and gives Himself up for us (5:1-2)–a topic that Paul comes back to soon afterward in the chapter, in the famous analogy between conjugal love and Christ’s sacrificial love for His bride! (5:22-33) The following chapter, immediately after this analogy, is certainly not destitute of practical examples of what it means for “the fruit of the light” to be “each {agathôsunê} and {dikaiosunê} and truth”, where those examples involve good and fair togetherness between persons.
So in at least three cases, maybe even all four, the term {agathôsunê} is definitely used in contexts that explicitly involve good-togetherness between persons, both between man and man, and between man and God.
{aschêmosunê} shows up only two times in the NT (so far as I can tell), once in Paul and once in RevJohn.
In RevJohn 16:15, the author translates Christ’s warning (echoing Christ’s warning to the Laodicians and to the Sardians earlier in Rev 3) that people will see the {aschêmosunê} of people who haven’t kept their garments clean.
In Rom 1:27, Paul is complaining about males with males bringing about {aschêmosunê}, having left the natural use of the female and being inflamed in their craving with one another instead.
Everyone agrees that the alpha at the beginning is a negative modifier, so the term means the opposite of {schêmosunê}. But what does that mean? It isn’t a term found in the NT. In the OT, the negative version of the term is a euphamism for genitals, so the cultural context is about nakedness in an embarrassing way (although usually the NT authors use a different term for nakedness, including in RevJohn and even in Rev 16:15 itself!) Plato is the earliest Greek witness to the term, which he treats as meaning a shameful impoliteness, and no doubt this is why Greek-writing Jews chose the term as a euphamism. And again, the immediate context of the two NT usages fits the euphamism well enough.
But what does the term itself mean? Notably, the OT usages, as well as Rom 1:27, tend to indicate that other people are the ones being negatively affected by what’s happening, not the ones doing the shameful thing (except maybe afterward in hindsight). The injunction not to build high steps to an altar of YHWH at the end of Exodus 20, for example, so that people will not see the genitals of whoever is climbing the stairs, involves those below being affronted. Similarly, no one could sanely interpret Rom 1:27 as meaning the two men are ashamed of what they are doing with each other’s genitals; it’s other people (including Paul) who are embarrassed for them. This leads to the interesting expectation that at Rev 16:15, it is other people, seeing the nakedness of those who haven’t kept their garments, who will be ashamed for those people.
Related terms in the NT, {aschêmoneô} and {aschêmôn}, tend to be used for indicating unpleasing shapes; which brings us to the question of the root meaning. The root term is {schêma-}, which is a variant of the primitive verb {echô}, to hold. The latter term came to mean much more than that, of course, but people arrived at those meanings as an application of the root meaning. {schêma-} itself applies the root meaning in regard to form or composition of an object. In English we would mean something similar by saying that an object holds a certain shape. Something {aschêma-} would, taken literally, be shapeless. A little less literally it means “ugly”. Yet {aschêmosunê} doesn’t mean ugliness, exactly. It’s a social breach where the importance is that other people are ashamed about it in an affrontive way (not the breecher, although he or she ought to be).
Another common concept of {schêma-} in the NT, derived from the notion of something holding its shape, is ‘outside appearance’. Sometimes this outside appearance is only a shell (like clothes, or a costume for an actor), sometimes it’s a proper form (like the human body) yet distinct conceptually from the essential reality of the form or {morphê-}. Thus a man who is truly a man can have both the {morphê-} and the {schêma-} of a man, but a supernatural creature (or even God) can manifest in the {schêma-} of a man (or anything else) while still not having the {morphê-} of what it is manifesting as.
(Or in the case of Christ, the Son can retain His original {morphê-} of God while also taking upon Himself not only the {schêma-} but the {morphê-} of humanity, as per the famous Christological hymn of Phil 2. That’s the distinction between the Incarnation of God, and a manifestation of God whether as a man or as a pillar of smoke and fire.)
So there are two senses of the word {aschêmosunê}, which overlap in their NT (and OT) usages, one usage of which overlaps with Greek culture usage generally. In one sense, clothes (which provide the outside appearance) aren’t holding together properly. In the other sense, someone has (accidentally or on purpose) done something that results in social structure or form not holding together.
Either way, the suffix happens to be just as appropriate as the prefix: the situations aren’t only talking about the lack of an abstract quality, but emphatically about a lack of togetherness (in a couple of different ways, one of which involves personal relationships.)
Relatedly, {euschêmosunê} occurs once in the NT, at 1 Cor 12:23, where Paul is drawing an analogy mixing the meaning of Gentile Greek usage for the term, social respectability, with the Jewish euphemistic usage for the negative {aschêmosunê}; and the context for the analogy entirely fits the concept of togetherness between people. We, the Christian body, are not one member but many, and not everyone can be an eye or even a foot. But that does not mean that we should disparage those who are weaker among us. “No, much moreso the members of the body which are regarded as inherently weaker are necessary”, and then Paul pushes the analogy to an outright scandalous level!–"and our {aschêmo} members”, referring by euphamism to the erogenous zones, “have more exceeding respectability {euschêmosunê}.”
It’s a daring play on words, but it wouldn’t work if {euschêmosunê} was not only regarded as referring to a good figure or external beauty, but also as referring to social decorum or propriety among people: because that’s the whole point of the surrounding context! Paul wants the Corinthian congregation to stop competing with one another and looking down on one another because someone has a gift that someone else doesn’t have, and start cooperating with one another instead in using those gifts together as a team. (In the 19th century, although I don’t know whether this is still true, some scholars thought Paul here for the first time coined the concept for “membership” in application to a group of people.)
{aphrosunê} shows up four times in the NT (so far as I can tell), three times in one area of a Pauline epistle, and once in a Gospel translated by a companion of Paul.
In Mark 7:22, it is found in a list of evil things proceeding out of persons defiling the person from whom it proceeds. The list definitely involves many examples of people not being well-disposed together with others; even the “togetherness” involved in sensuality (literally “leading while the moon is absent”) or fornication involves exploiting other persons (or even degrading one’s own person-ness).
In 2 Cor 11:1, 17, and 21, St. Paul is sarcastically talking about how the Corinthians considered him not disposed together with them in his previous boasting at some time prior to the writing of 2 Cor. (The first two parts of the term definitely mean “not-disposed”, and the context of the whole passage is about people described this way not being disposed together with the Corinthians, whether actually or in the Cors’ imagination.) Part of this sarcasm involves pointing out that they have no problem accepting other people who come to them speaking {aphrosunê} and are quite patient with them! (Compare with verses 19-20, “For you are bearing with it if anyone is enslaving you, if anyone is devouring [you], if anyone is taking [from you], if anyone is elevating [himself over you], if anyone is lashing you in the face!”) Whether he is continuing to be sarcastic or not at verse 17 (although I think he is), his claim of {aphrosunê} in his speaking is explicitly based on not speaking in cooperation with the Lord. (The sarcasm I understand to be that since He does speak in accord with the Lord, then it wasn’t {aphrosunê} after all.) Paul seems to be saying in verse 21 that (sarcastically) he is speaking {aphrosunê} to complain that by such behaviors of other people behaving {aphrosunê} toward them (exploiting them instead of being in wholesome togetherness with them) “we are weakened”. The whole context is about {aphrosunê} involving a togetherness of people in exploitation of one another (and Paul sarcastically defending himself from the charge that he is behaving in such a way as well.)
One of the regular translations, “foolishness”, is very weak for the kind of behavior being talked about here: someone who Paul considers really speaking {aphrosunê} to them, is charged by Paul with crimes against their humanity!–and Mark’s application of the term in translating the warning from Jesus, occurs in a list of similar crimes against humanity! “Foolishness” probably comes from the root middle being a metaphor for the heart and therefore a metaphor for the mind (although literally it is a modified version of “rein” or “control”). “Heartlessness” would be a better translation for the contextual strength of the term’s occurrences: exploitative evil rather than silliness or stupidity or even irresponsibility.
{eleêmosunê} occurs several times in the NT, always in the connotation of doing a good deed for another person (typically involving giving money to the poor). Matt 6:2-4 (it is better to do mercy for people in secret than to do it for public glory); Luke 11:41 (Jesus instructs the Pharisees to give what is within them as mercy to people); Luke 12:33 (Jesus instructs the rich young synagogue ruler to sell everything and give the money as mercy to people); Acts 3 (the story of Peter and John healing the lame beggar, giving mercy to him in this way rather than giving him money); Acts 9:36 (Tabitha famous for giving mercy to people); Acts 10 (Cornelius the righteous God-fearing centurion gives mercy to people regularly, as an object lesson to Peter); and Acts 24:17 (Paul was arrested and handed over to the Romans while delivering the donations to the Temple, which he had collected in Asia for giving mercy to the poor).
It’s hard to have mercy without a person as the object anyway, of course; but the point to the term seems to be an emphasis beyond the attitude of mercifulness to practical action for other people. Mercy-togetherness is not a bad translation of the concept (though it might be rather unwieldy in English).
{hierôsunê} occurs three or four times in the NT (depending on the text transmission), all of them in chapter 7 of the Epistle to the Hebrews, as a noun for the adjective “sacred”, and is specifically deployed as a metaphorical way of describing priesthood and its duties. (It’s a loan word from Greek going back as far as Plato, and is somewhat more frequent in the Jewish Greek intertestamental texts like the Maccabees. The Hebraist may be using it because he’s building his analysis of Christ’s superiority in comparison and contrast to Philonic Judaism, which followed Philo’s emphasis on Moses, angels, Melchezidek, etc. being presented in terms of Jewish Platonism.)
The Hebraist’s argument here is that the Messiah’s priesthood is superior to that of the Levites, thus explaining why the Son of David was promised to be made a priest despite coming from the line of Judah instead of from Levi (a point which led sometime in the first century, before or more likely after the time of Jesus, to the theory of two Messiahs among Jewish non-Christians: a priestly Messiah, who would probably be the one to suffer and die, and a King Messiah from the Son of David, who would avenge the death of the priestly Messiah and maybe resurrect him! There was much rabbinic debate about this afterward.) The Hebraist takes the solution to be that David’s Son was promised the priesthood of Melchizedek, of which the priesthood of Levi and his sons was an inferior figure since Levi through his ancestor Abraham gave religious tithes to Melch. (And then Melch himself was a figure of the Messianic priesthood to come, a priesthood conferred directly by God, as indeed the Levi priesthood also was, but for the Messiah also a priesthood through inheritance from God, prefigured in the inheritance of the priesthood by the descendents of Levi.)
The first two (or three) instances of the term seem neutral to the question of whether {hierôsunê} involves anyone or anything being holy-together; but the fourth and last instance at Hebrews 7:24 leads immediately into an explanation of what is most important about Christ’s {hierôsunê} (and about anyone else’s {hierôsunê} for that matter in any contemporary Mediterranean religion, whether pagan or Jewish): because Christ’s priesthood remains inviolate “into/for the eon”, He is able to save (from sin) into the uttermost those who are coming to God through Him Who is always living to be pleading for their sake. (Unlike Levitical priests who die and must be replaced.)
The point, which the Hebraist goes on to discuss further in chapter 8, is that Christ acts as the ultimate possible mediator between God and man, bringing sinners to be sacred together with God. This special emphasis on salvation from sin, in the Hebraist’s argument about why we should regard the utter supremacy of Christ as priest to be so important, might be why he chose (and/or was inspired) to use the term {hierôsunê} when he could have used {hierateia} (as in fact he does in the same chapter 7:5 for the priests of Levi) or {hierateuma} (“sacred effect”, as Peter prefers in 1 Peter. To be fair, the New Testament doesn’t talk much about priesthood by a term we would translate into English as such in any case.)
{megalôsunê} occurs nowhere except in Greek Judeo-Christian texts, and only three times in the NT (but often in the LXX). It’s a Jewish euphamism for God or for divinity, and two of the three NT occurrences directly follow suit.
The exception is Jude 1:25, where the author closes the letter with a standard doxology “to the only God, our Savior, through Jesus Christ our Lord: glory, {megalôsunê}, might and authority, before the entire eon, now, as well as into all the eons, Amen!” This occurrence wouldn’t necessarily have to involve anything being “together” in some way–although trinitarians might find the term usage interesting, especially after reading the other two NT occurrences!
At Heb 1:3, the Hebraist declares that the Son, being the result of the radiance of the glory (or shekinah) of the God and the emblem of the hypostasis of the God, besides carrying or upholding the All by His powerful declaration and making a cleansing of the sins, is seated in the right (i.e. in the right hand) of the {megalôsunê}in heights, becoming as much better than the angels as the allotment of the more excellent Name He enjoys (than the angels). Megalôsunê is here obviously a name-title for God, as from the Greek OT. But even a unitarian would have to acknowledge that this greatness was being shared together with the Son in a way that is just not true for even angels (which on any interpretation is practically the whole point of the Hebraist in the first chapter); much moreso would a trinitarian affirm this! “Great-togetherness” makes perfect contextual sense here, especially (but not even exclusively) if trinitarian theism is true.
The Hebraist is making much the same point (whatever the Christological implications are otherwise supposed to be), at Heb 8:1 (a portion of scripture already discussed in this analysis for other reasons): the sum of what he has been talking about for a while is that the importance of Christ as our Chief Priest is that He is seated in the right of the throne of the {megalôsunê} in the heavens, from which He acts as a minister of the holy ones and of the true tabernacle, which YHWH pitches, and not a man. (Whether “and not a man” is supposed to refer to “which YHWH [and not a man] pitches” or to Jesus “[and not a man] acting as a minister of such and such”, is another debate, but I’ve punctuated it so it can be read either way.) Megalôsunê again refers to God Most High and the contextual point is that the Son shares an importance with the Father unique to God Most High which affects the quality and effectiveness of His chief priesthood for us. “Great-togetherness” once again makes perfect contextual sense.
{sôphrosunê} occurs three times in the NT (both from Paul). It starts in non-Jewish Greek as far back as Homer, and generally connotes sanity against mania, thus soundness of mind. In English we would say it means someone has their wits together instead of scattered. From this meaning derives a closely related notion of being soberminded or self-restrained.
Paul’s advice to Timothy (one verse of which might be a gloss to Paul’s text by Timothy himself) at 1 Tim 2:9 and 2:15 fits the meaning of sobriety and self-restraint compared to wild behavior: women should dress soberly, not wantonly (in effect they should keep their clothes together and not flaunt themselves by wearing strips of clothes–the term itself is practically a pun for “whole-midriff-together”!), and should continue (bearing children?) in faith and love and sanctity with self-restraint.
Paul’s defense before Festus in Acts 26:25, on the other hand, more directly involves the opposite of insanity: he protests that he is not out of his mind but uttering words of {sôphrosunê}.
The root word {sôphrôn} is in any case recognized to be itself a compound of the term for safe/save {sôzô} combined with the term {phrên} for middle, which in this context refers to the fact that the Biblical authors (with their time and culture) thought the heart, as the obvious center of the person and of the person’s life, was also the center of the person’s mentality, home of the will. (Although “middle” could also refer to the feelings of the person by reference to the midriff of guts where they believed the emotions came from.) There are several cognates of {sôphrosunê} in the NT (and non-Judeo-Christian Greek), and they all involve this same concept of reliably coherent mind and/or feelings. The mind or the feelings aren’t scattered or shredded. They are coherent; they hold together.
An emphatic “Safe-mind/feelings/heart-together” turns out to be an entirely appropriate translation of {sôphrosunê}.
) “The good” in nominative form would still be {ho agathos}. This doesn’t negate what you report about qualitative abstract nouns, but I thought I should mention it in passing.
) as I am when it comes to Greek, I’m open to the possibility that you will find all such evidence, in which case I’ll need to discuss the implications of “righteousness” meaning “just-togetherness.” Let’s see if we have to cross that bridge.