You should feel fortunate that your H wants to work on this. So many of us don't even have that. You are already so much further in repairing your marriage than most of us are, and you could blow it any second with your negative thinking, snooping, and distrust.

You have two little girls who need you more than you need your H right now. Focus on them, on yourself, and stop worrying about what he's doing. He's told you all of the things you needed to hear (he's over the OW, he loves you, he wants to work things out, he's working for YOU and the kids), and yet it's STILL NOT GOOD ENOUGH!? What will be enough for you? Does he have to time-travel back and undo the affair? It happened. He apologized. Get over it. And like so many have already said -- you can't ever know if he'll do it again. But you can't know that about ANYONE, because you can't ever control other people's actions. You can only control your own. So start right now and learn to react with more positivity and trust toward your H -- I bet you'll find a better R on the other side if you just give in to the good feelings instead of letting the bad ones control you.