I am so sorry you are going through this, what a shit to have an affair when you have small children.
My father had an affair when I was little, DM had given up everything for him. She was happy and loved him a lot, she thought everything was perfect, 5 years later we found out. I remember that time well, I was 8ish, us kids had no idea what was going on, except there was lots of shouting, lots of crying, lots of apologies from our dad.
They stayed together, my DM cried a lot, she got herself a degree and career etc.. now I have kids I just can not understand why, when she needed him the most, when we needed our dad he was off fucking some other woman.
And then joy of joys, it turns out he didn't stop seeing her at all, years later they had picked up where they left off and DM found out again, by accident. This time it nearly did break them up, but now she feels too old to start over and be alone. They have 5 years down the line seem to have sorted stuff out, they finally seem ok, DF is very attentive to DM but really what life has my DM had that is real, and that resentment now bubbles under the surface.
Please don't stay with him for the kids, don't stay with him because you are scared, you are young enough to turn it all around, to be happy, truly happy, to be able to trust again you deserve that, your kids deserve that.
Of course it is shit now, but why are you the one trying to save the marriage, finding the kids too much, what a twat, not too much for you to manage though on your own! Having small kids it tough on any marriage but this is the time when you need each other most, and he wasn't there for you, so do you want him in your life anymore? I think it is he that needs to demonstrate to you how sorry he is and how it will never happen again and you need time to think about what you want from life. Good Luck, and I am sorry you and your kids are going through this shit.