Nazarite vows could go different ways, giving up alcohol wouldn’t have to be one of them. Nazarites normally didn’t go bald, but rather didn’t cut all or some part of their hair–the root of the word itself comes from the description of an unshorn vine, and the first institution of Nazarite vows (in Numbers 6) primarily involves this. (The Turin Shroud shows a bit of a well-kept tail haircut.)
It’s just a visibly enacted self-disciplinary faithfulness as a witness to other people that God is worth making sacrifices for. A vow of chastity would serve the same purpose, only moreso.
Notably, while Jesus made water into wine at Cana, I don’t recall that we actually see Him drink wine until the Lord’s Supper, at which time He takes a vow not to drink it again until He can drink it anew with them in the kingdom of God to come. And it’s kind of fuzzy whether He Himself actually drank the wine even then! ("‘Take this, and share it among yourselves; for I am telling you, I will not drink of the fruit of the vine from now on–until the day I drink it anew, when the kingdom of God arrives.’ And all of them drank from the cup.") I doubt a vow about this was supposed to exclude a Jew from partaking in the Passover ritual, though, and that required wine. Strictly speaking we only see Jesus drink wine once: the sour wine offered on the cross shortly before He died, finishing His sacrifice.
If a Nazarite broke his vow or became unclean in any way, He was required to offer a special sacrifice and start the vow over again. The hair was originally grown long (in the original criteria established in Numbers), but was intended to be cut off eventually to be offered as a burnt sacrifice to God, when the vow expired. (For this reason King Agrippa and his brother-in-law King Jannai each sponsored half of a group of poor Nazarites who had come to complete their vow by shaving their heads.) If the vow involved all the hair on their heads, then all the hair would be shorn at the completion of the vow, and that may be where you’re getting the idea of bald heads.
The duration by the day of Jesus was voluntary and could range from an hour to a lifetime, set to end under various conditions specified during the oath. In the days of the Mishna compilation, Nazarite law filled a whole tractate, and the standard length was 30 days, but two or three oaths were usually made at once each regarded as separate to be fulfilled one after the other.
While the original version of Nazarite vows required that they not touch corpses, Samson Nazarites were allowed to do so in honor of the heroic Nazarite judge. (On the other hand they weren’t allowed to cut their hair for any reason, while normally perpetual Nazarites were allowed to cut and sacrifice their hair once a year to keep it from being unruly. Note that Samson engaged in drinking bouts and certainly had an active sex life!–though this was not necessarily a good thing. 
) In the case of Jesus, He brought corpses back to life by touching them!–so it is debatable whether He would have been rendered unclean by them. (Worth noting though is that after healing His first leper, whom He had warned not to tell anyone about but who went out telling anyway, angering Jesus, He was required to stay outside towns until such time as they would expect Him to perform cleansing rituals! (Mark 1:45 & par) That He’s later easily shown entering and leaving cities despite His increasing popularity indicates the problem wasn’t popularity but perceived contamination from touching the leper.)
Note: some of the preceding information is borrowed from the Nazarite entry of the 1908 Jewish Encyclopedia, which can be found here. The information is admittedly not inerrant, as the editor mistakes John the Baptist for Jesus once.
I’m both impressed and scandalized by everyone’s humble openness to this possibility. Wow.
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