Greetings MotherOfTwoJRF,
I am truly sorry for this trial you are enduring. Unfortunately, though well intended, believers usually make this worse by badly using Scripture. (not accusing anyone here of that, just saying) For this reason i don't use Scripture in these situations.
First thing is pray. Pray for yourself first, then your kids and then for Him. When you pray for yourself Ask for healing and wisdom do proceed as God would have you go. For the kids, that the Lord act as a shield to protect them. For him, forgiveness and for hurt. Let me explain that last one more.
When I say pray for forgiveness its for 2 kinds of forgiveness. 1st, is that the Lord will forgive his sin, intercessory prayer. 2nd is that he can forgive himself. This means he will first have to admit that what he did is a sin against you and God.
Now to the hurt. I said hurt not harm. People that indulge in ungodly sexual activity will never change for someone else. At least not completely. The thing is, until they themselves hurt enough, they will simply not seek out God and the healing and life change He can give. So pray that he hurt enough to seek out the Lord's forgiveness and His help in healing from this behavior.
If he does come to this point I am going to tell you the cold hard truth, its temporary. Unless he is accountable to someone, preferably another Christian man or men, the odds are sorely stack against him. Despite popular belief we forget nothing, ever. We simply have a hard time recalling it but its all there. Ever be doing something and a memory just pops up? well all the things he has done will at some point pop up too and if he doesn't have the support in place, falling become more and more likely.
Just search your local churches for men's sexual addiction and accountablity groups. Also it wouldn't hurt to get in a Christian womens support group that does not bash the addict. They can help you to heal and to pray and to forgive. Good Christian based counseling wouldn't hurt you both either.
I hope this helps you find your way out of the distress you are in.
In Christ,
Bishop SEH