It really doesn't matter if you have a "drinking problem" or are an alcoholic. In either case, it's best not to drink. Anxiety and depression are common when one stops drinking. In fact, both are commonly found in alcoholics whether they drink or not. I've been sober for a few years now and take an anxiety med when needed and an anti depressant daily. Many of my friends in AA are in the same boat. So it's no big deal to us. I need to mention, though, that I was prescribed Lexapro years back and had some very bad things happen. So when you start taking it (and you should try it) just be careful. If you start to weird out stop immediately and have your doctor put you on something else.
You won't suffer any damage from the side effects if you don't continue to take it
Yeah, yeah, I know the policy about talking about meds, but that was important.
And let me echo Golfman. Stick around. Perhaps take in some AA meetings. You'll be all the better for the experience.