Which leads to me to question whether or not I was using alcohol to fix my depression and anxiety
As others have said, it's not an either/or proposition. It's more like both/and.
YES, alcohol has some positive effects on mood and anxiety, so some anxious and depressed people can sometimes find some temporary relief by drinking.
BUT, those same people can also be alcoholics, where they want to control their drinking but find themselves unable to do so. (Or pick another definition of what "alcoholic" means... there are several good ones out there.)
The trap is to think they are mutually exclusive, namely the false idea that if drinking was primarily a way to control anxiety and depression, then when the anxiety and depression are cured, problem drinking will turn into manageable drinking. I think for most people that's simply not going to be true.