A lot of doctors may dismiss it, but I wouldn't. Talk to more doctors if you can afford it. It's pretty rare to have "natural" stuttering start at adulthood, and the fact that it coincides with alcohol withdrawal is suspicious. It sounds like it could be a symptom of something else... which means you should find out what that something else is, since it could be important, and it also means it might be treatable. Unfortunately, a lot of doctors can be pretty dismissive about things they don't understand, or if they have certain biases (like if the patient is black, a woman, overweight, an alcoholic, etc etc etc...). So, get a second opinion, or a third. Definitely find a neurologist.