Tpen,
I am with you and can only tell you that I have had those same struggles but they will lessen with time. She was a big part of your life and despite the cheating and alcohol, you loved her anyway and you got used to her being a part of your life, and thought she would always be there. It is a major life change and a big adjustment.
And likewise, it is hard to fully let go and accept the fact that your relationship is over. But it may very well be. In fact, the present fact is, your relationship is over. And even if she is allowed back into your life (and this is all YOUR decision, not hers, should that chance come), your relationship will never ever be the same again. It is changed forever.
Still, I think the hard truth I have had to face in my situation, and you face in yours, is that our exes find it easier to run from the truth than face it. In getting into a new relationship, it's a "fresh start" to them, one without guilt, admission of problems, a person who has not seen their darkest sides, a person they do not have to look at and feel the guilt over what they have done to you. It, much like the alcohol, is their escape.
What works in your favor, with NC, aside from giving you a break, is that they WILL remember the fond happy times in your relationship as time goes on, probably even right now. Memories are strong. Unless you fought all the time, and I don't think you did, the new person does not have the benefit of the life you led together. Sure, they can make new memories, and in time those may be fonder to her than with you (yes, it happens unfortunately) but the idea of NC is that by then, you will feel the same way and be moving on in your life as well.
Very tough times right now, as it's hard to let go of them, and the past, and imagine a future of never seeing them again, but it is a reality we must face, at least for today.
Stay strong.