I'm relatively new to this but my plan is to not tell hardly anyone unless I can help it. And I definitely wouldn't label myself as an alcoholic. I don't care if that's what it is, it's none of anyone's business just like any other medical issue I might have.
If I do have to tell someone I'll just say that I have a problem and can't drink. That's all they need to know.
As far as the doctor goes if they ask I'll tell them the same thing, that I had a drinking problem in the past so I don't drink any more.