Ria, I think he does want to be with you and your children but as others are saying, he wants it all and on his terms. That would be fine if you trusted him and were not worried about his other circumstances but in your own words, this situation is killing you and you're not the person you used to be.
He would have to make an almighty gesture to try and make things up to you but he doesn't sound like the type who can or will.
You ask how you can be happy with your kids if you move on.
You're not happy now plus you have lost your confidence.
If he truly can make the commitment you want and change his behaviour, fantastic, and absolutely wish you the best.
But if not, you deserve better, at least you might have a chance for happiness and restore your confidence if you move on. But if things don't change big time at home, you need to take control and be in charge of your happiness and self esteem, for your sake and sake of your girls.
I know, it's easier said than done to just leave him or kick him out. Hope you can find the courage to spell out to him what you want, take it step by step.