Well good luck. Just keep in mind a few things:
A DUI requires at least one year from the date of conviction and a waiver from the recruiter before you can enlist.
If you continue drinking in the military and get a DUI that means bad stuff, like there is a good chance of a dishonorable or less than honorable discharge depending on the circumstance.
I would solve the alcohol problems before you join the military if you can get in, you will have some time to do that, I would also start doing things which demonstrate that you have rehabilitated yourself that you can prove when you fill out the paperwork for your waiver for enlistment. DUI precludes you from AF, Coast Guard, and Marines unfortunately. Enlisting in the fall will give you a better chance at getting a waiver as the recruiters get a flood of new guys coming out of HS in the summer/spring time.
Good luck. If you are an alcoholic drinking some of the time will eventually not work. But that is something one must find out on their own, it doesn't sound like you have hit bottom yet.