Drinking can't cause stuttering. For some people, it can make stuttering worse though. Others stutter less when drunk. Also, slurring aside, "tripping up" over words is 'better' than blocks and can be better than prolongations as well as there is less holding back. I'd suggest finding a good speech therapist who understands stuttering to help you deal with your own shame because what's making it so difficult for you to listen to.