Op, this is difficult, because it sounds like you're both staying in the marriage because you prefer that to going it alone.
And that's ok. But you need to be honest with each other about why you're trying to make it work, and come to some form of agreement on the way forward.
As for telling your kids, you don't need to do this if it makes your life harder. It's not about protecting him, it's about protecting yourself.
However I'd echo what pp have said, you're already living apart. Think long and hard about what you want for your future. And why you want it. Then talk to him and be honest.
Do you want romantic love, faithfulness, togetherness? Or do uou wish companionship and to continue as is? To maybe have an open marriage?
He was clearly willing to leave the marriage for this woman, and is treating uou like a companion he can he can off load on.
Think of the terms you wish to continue under and then lay it out for him.