Right, I want to give an opinion here;
First off, my own reflections: I've been (falsely) accused of cheating, it sucks and it almost tore my relationship apart. I've often wondered what I'd do in the given situation that my girlfriend cheated on me, and I've been on both sides whether to dump her straight away or try to sort things out. The situation came true when I was told that she actually did (turned out to be false, but still) and I was in no doubt whatsoever that I was better off by myself. Cut all contact, and I never wanted to see her again.
To clarify a bit, that particular relationship ended by other means, IE there was no cheating involved, but now I know how I feel on the subject.
On to yours, Vanguard, I would say that you should break it off. It hurts now, and it will hurt later on. I still miss my ex six months after we broke up. However, few can live with the niggling doubt in the back of their heads. That fear of having to go through an unfaithful significant other will make a healthy relationship incredibly difficult, arguably impossible.
I've always had the thought that being single is way better than being in an unhealthy relationship, and considering the long distance between you, I honestly think you're much better off until you find yourself someone better. Someone who's fully committed to you, and you alone. Alcohol is never, ever and excuse.
Also, I'd like to know that you've got my most sincere sympathies. Hope your spirits will raise soon. I know I'm not one for relationship advice, but I hope I've at least shed some light on something.
Good luck, chap.