When you're 20 you care what everyone thinks, when you're 40 you stop caring what everyone thinks, when you're 60 you realize no one was ever thinking about you in the first place.
Seriously, though, this quote is true. Not sure how old you are, but I'm approaching forty. What he says stands up to my experience pretty well so far.
If someone offers you a drink, you can politely decline, and they won't think anything of it, most likely. If they persist, you can tell them you're not drinking, and they'll probably not think anything of it. If they STILL persist and ask why, you can tell them the facts: you're trying to eliminate drinking, etc. Just don't get peachy, and it'll be fine.
At that point, if anyone makes more than moderate amounts of fun of you or the like, they are pretty much a douche and their opinion really shouldn't matter to you anyway.
I think you'll find, however, that most people will accept it and move on. Trust me, they are not worried about you. Like you, right now, in this post, they are mostly worried about themselves and their problems. You might find more support and positivity than you expected, too. :)