From own experience I would say that stuttering is not due to social anxiety, rather it is increased by it. I do not see myself as a socially anxious person, in most situations, and I stutter anyway. I have also read a lot of studies made(available to you on google) on how different medications affect stuttering. And some anti-anxiety medications work on some people, however it all comes with side effects. The only drug I have tested is beta blockers, which don't effect the brain, but the heart rate. It calms you down and you get focused. Did make me calmer but did not reduce my stuttering. I honestly don't think a medicine can solve the whole problem, maybe make it a little easier. I am however a strong advocate of practicing speech technique, using hearing aid which echoes your speech and simply boosting your confidence, instead of using medication. I feel THAT has worked best for me.