Hey there. I'm so glad that you found us for support. I'm sorry that you have the reason to be here.
You love an addict. That's the bare bones of your situation. Your WBF is an addict. An addict actively in their addiction will always choose their addiction over you, your children, and anything else in his life. It doesn't matter if you love him, if he loves you, if he loves his children, if he loves God, or anything else that is sacred to him.
An addict who is in the active phase of their addiction will always, always choose their addiction. And anything that allows them to continue in their addiction. You cannot love him out of it, you cannot guilt him out of it, you have NO control over him or his addiction.
The only control that you do have, is in you your reactions and your responses. This is going to sound harsh, but if you can drop your emotions for one moment and take a cool, long, clear-headed look at the past and what that means for the future, you're going to see that you are in a cycle of rinse, lather, and repeat. And you are going to stay in that cycle until you break free, with very little change except possibly adding more children to the cycle children that you bear and children that he fathers on the OW.
Please. For your sake and for the sake of your children, see a lawyer, file for CS for your children so they are first in line, and show him the door. Let him know that until he is clean and sober for a year, he cannot come home. That you need actions showing that he can be consistent in his sobriety. And take care of yourself and your children.
Yeah, it's likely that he's going to crash. But he's crashing already. Yeah, he might go crawling to the OW. But he's already doing that. Please understand that no matter how brutal this sounds, he's already left you. He's already made a choice. He cannot and will not get sober and maintain that sobriety as long as he can Ping-Pong between your house and her house. As hard as it is, you need to pull out of the equation for your and for your children's sake.
(((hugs))) I'm so sorry. I know this is just brutal. I know this isn't want you wanted for your life and for your children's lives. But you're the only person that can take control, for yourself and for your children. Please come back often for support. Everyone here wants to help and lord knows, we've all walked, and are walking, similar paths.