Well there's not a "legal" description of alcoholism, and the "disease" of alcoholism even existing is still often debated. I think what it comes down to is what has resulted as a consequence of said 'alcoholism' in your life.. missed responsibilities, lack of dependability, inappropriate intoxication, etc etc (not that this is you, just a few things that can happen as a result of drinking too much), not to mention risk of DUI and other horrible decisions.
I think your actions need to prove your ability to be a good father, and I also agree that the best tactic in protecting yourself legally will be a good lawyer. I hae also heard that joint custody is very rare.. often there needs to be a primary custodial parent, and a parent who gets visitation, as the kids have a lot more stability that way (in most cases), but I have no idea what will happen in your case.
Every alcoholic I know has called themselves a 'high functioning alcoholic'.. ah the oxymoron of addiction..
From what you describe, I think regardless of the kiddos involved, your best bet is to pursue recovery and lifelong abstinence from alcohol, you've put yourself (and potentially the kids, if even the babysitter wouldn't release them to your care) in some scary situations whilst drinking, and thank goodness that's all over with!!
I wish you well on your road to recovery.