I gave up the drink about 7 years ago now, though had a couple of relapses, but continuously sober now coming five years and the best thing I ever did. For years I wouldn't class myself as an alcoholic due to the stigma mainly, I always looked at other drinkers who were worse than me and concinced myself I wasn't alcoholic.
The first step is confronting and admitting the issue, you've done that here which is a big thing. The next step is to ask genuinely whether you are an alcoholic or a problem drinker. The thing I have picked up on in your post is when you say 'I don't think I am an alcoholic ...', by saying 'I don't think' means that you don't know, although at this stage you see yourself as more of a problem drinker.
For either issue alcoholism v problem drinking there are solutions. For the first there is AA or some other program aimed at alcoholics, for the latter there maybe counselling, or educate yourself on the effects of drinking to your body. Either way it is best to approach with an open mind, maybe even try an AA meeting and if it's not for you then try something else such as the medical profession, or counselling.
It took me a long time to figure out why I drank, it also took me a long time to admit I'd crossed the line to becoming an alcoholic ... I was never the park bench type, I never drank in the morning and never drank vodka, never went to prison, never had medical alcohol issues, never lost my licence, therefore it was easy to convince myself that I was just a problem drinker.
I am not saying that you are an alcoholic, I don't know, but picking up on the fact that you are not sure tends to mean that you have thought along those lines at some point yourself. If moderate drinking isn't working it maybe an idea to have an open mind on the issue. Either way, problem drinking or an alcoholic there are solutions, which start with confronting the issue head on.