Drinking for the sole purpose of getting drunk isn't drinking "socially"
In order to keep your drinking to just socially, maybe you should discuss your poor sleeping ability with your doctor and get a mild sleeping pill to help you with that? Of course talk to him about anything he gives you to make sure it's not addictive. At this point you're not sure if you have an addictive personality or not so be cautious.
If you can't stop drinking in order to get drunk, then I would say that, yes, you may be headed for a drinking problem/addiction. Personally, I'm not sure how you cope (even occasionally) getting drunk and then having to get up early to tend to your wee one. Blech!
“The willingness to accept responsibility for one’s own life is the source from which self-respect springs.” ~Joan Didion