Sorry to say, I don't think so.
don't worry about snooping, because it is only bad snooping when it is none of your business, when you have evidence, it is being proactive and protecting yourself and your childrens best interest.
I agree with Melissa, do you want to have to worry and wonder and live like this all the time?
block all internet from him.
He wants/has his cake and to eat it too.
He has his "thing" on the side, and doesn't have to pay anything to you, to her, or to the kids in support.
If you truly want to make this work, get him in counseling now. No excuses from him and block him from the internet and make him get rid of his pay as you go phones.
Sorry just re-read you are doing couseling, maybe add talking a pastor together.
part could be just the addiction to the internet and he just happened to find a bite, or he may just want the cake thing. Just realize, he is going to say as little or as much and whatever you want to hear for him to stay, until he decides he wants to go.
I would still be watching your accounts, and make your own NOW. Start your credit NOW if you don't have any. Make sure he isn't taking any out and preparing for something. Just to be on the safe side and not get blindsided.
For me personally, he would be out already. If I couldn't afford the house, it would be for sale with me in it, not him. If I chose to give us another try, he would have to earn his way back in.
hugs and sorry again