You sound scared, desperate, and a little in denial. No judgment from me; I understand. I also understand why some posters are frustrated with you. It’s blatantly obvious to everyone here that this man is an utter shit to you, and that you NEED to leave, both for yourself and for your children.
You don’t need proof. You don’t need him to actually stick his dick in someone else before you can leave. You need to recognise that you’ve already got proof that he’s crossed a line - he’s pursuing another woman and you, probably accurately, believe that he wants to make this physical.
You say he’s “not really abusive” any more, but what do you mean by not really. Is he abusive but it’s not as bad as it used to be, so you’re figuring that’s a good thing? No abuse is okay!
Counselling isn’t going to change him, but you might benefit from seeing someone on your own. It sounds like your self-esteem is at rock bottom, and you really need to realise that you are worth SO MUCH more than this!
Secretly getting a mortgage isn’t going to get you out. You need to find somewhere to rent, or move in with a friend or family member in the short term. A women’s refuge might be a terrifying idea, but even that would be a better situation than the one you’re in now.
Do the Freedom Programme online now; you can try to meet other women in the summer when you have more time, but you can make a start immediately.
My situation was different because I didn’t have children with the dickhead, but I’ve been in an abusive relationship. I too spent almost three years believing that I could change him, that he did actually care about me (I wanted to believe he was struggling with personal demons or something 
). He was a master of gaslighting and I thought I was to blame for many of his actions. He moved onto threats of physical violence, but led me to believe that if he hurt me it’d be my fault - he couldn’t be held responsible if I had wound him up to the point that he hurt me. Throughout the relationship I had suspicions he was seeing someone (or more than one person) beside me, and I saw evidence of this, but he accused me of having trust issues and made me believe I was paranoid, had mental health problems and was imagining it all. He tried to isolate me from my friends (and nearly succeeded), he would create a drama whenever I saw them and was very vocal about how awful they were for me.
When I eventually ended it, purely because I’d had enough (he threatened violence against my male friends, turned up at a night out I was at and demanded I leave, aggressively swore at me), he tried to love bomb and win me back. I told my female friends I’d left him and their response was to say “OH THANK FUCK”, hug me, then take me out to celebrate.
He carried on trying to win me back, sent pleading messages saying how much he missed me. I then found out he had been simultaneously seeing at least 8 other women the entire time we were together, I was actually the OW when we got together, and he’d been in an apparently serious relationship for three months before I left him. The other women and I ended up becoming friends (granted, that’s unusual!), I had counselling, I can see him for the utterly pathetic specimen he was and my life is SO MUCH better now.
I’m so sorry for writing such a long response, and for going into so much detail!
You deserve better than this, and ripping the plaster off and getting out would be the best thing you could do. Everyone on this thread is behind you, and I’m sure everyone in real life would be overjoyed if you left him too. You are stronger than you realise and you can do this!