I don't know that I'm a full-fledged alcoholic yet or whatever since I haven't been drinking as long as probably a lot of people.
Only you have the right to call yourself an alcoholic; but, it does sound like you've passed all the "tests"...and, no, you're not too young to come to the realization.
My suggestion would be to go to AA meetings for the next three months...of course, while you're not drinking (if possible). By the end of that time, you should have established a good network of support and an excellent program of recovery. You should also be more comfortable in accepting alcoholism as an "equal opportunity disease" that doesn't discriminate...there are all ages, ethnicity, economic status, etc. of alcoholics.