Hello and welcome. I'm glad you're here, but sorry you are in position to need this kind of support.
Before anything else, I just want to let you know that I had a hard time with the idea that there is a difference between "giving up" and "letting go."
But that's getting way ahead of ourselves here. First and foremost, it's imperative that you a) educate yourself about alcoholism, and b) build a support network for you (online here at SR is a great place to start, but face to face, real world support is invaluable as well).
The first thing I was told about alcoholism were the three C's: I didn't Cause it, I can't Control it, and I can't Cure it. This disease is entirely your husband's problem to deal with. It's not one that you can wish away or ignore until it gets better on its own.
Of course that does not mean that you and your children are not profoundly affected by living with an active addict in the house. You have already felt the financial burdens of the disease, and the emotional ones. It sounds like you and your kids are living in a minefield. You'll never be able to dig them all up on your own, and until your husband recognizes the problem and decides to seek help of his own free will, the best you will ever be able to do is manage.
When you say that cutting your losses goes against everything in you, can you elaborate on that some more? Where does that come from?