(P.S. There is ABSOLUTELY NO WAY in my profession I can say "I am a recovering alcoholic".)
Isn't it sad, that in this so-called "enlightened age", there is still such a stigma attached to the word alcoholic? But, it is what it is. Personally, I think people fear the word because they're afraid they might have to take a long, hard look at themselves and their own drinking.
My only suggestion would be to keep it simple: "Thanks, but I just don't feel like drinking tonight." If they persist, "Maybe later....but, thanks, anyway." I bet...unless they're a bunch of insensitive idiots...they'll eventually either get the idea, or just quit offering.