I experienced social anxiety about 4 weeks into sobriety. Six months later, I was dealing with it more often. Usually in public speaking situations when I would start sweating and hyper-ventilating.
I considered seeing a doctor for it but decided to manage it with regular exercise and cutting down coffee on days when I was due to present publicly. It went away after a couple of years of sobriety but I did have to manage it quite actively.
I realised that I'd masked my natural shyness with drinking and for a time wondered if anxiety had just been latent in me all along. With more time having passed, I don't think I actually have anxiety. I feel perfectly fine now and haven't had an anxiety attack for about 2 years. I really think what happened is that I messed up my nervous system and chemistry with prolonged heavy drinking.
This is just my personal experience. Certainly not meaning to discourage you from medication if it helps your situation.