ItIf you have no specific medical condition that would be exacerbated by anti-anxiety drugs, I'd say you should at least try them to see if they work well for you. Most of them don't have serious side effects if you don't take too many, or too often.
Excessive drinking has well-documented negative effects, among them the possibility of violent behaviour that can get you into really bad trouble (as in: gigantic fines/damages, or even jail time) on a plane.