He will have his own narrative, and it may paint you as horrible - not caring so feeling happy, having been lying to him for years, whatever. It's his defence mechanism as he does not want it to be his fault. Not that he's sitting down and working out a story; it's just his brain furiously trying to make sense out of what is going on in a manner that protects his self-esteem. Our brains are good at that. That's how you got into this position.
I know it's hard, but when he comes up with something that feels like a slap in the face, try to remind yourself that it's not about you, it's him working on his defence.
If you find the right counsellor, it can be hugely helpful. I did counselling after my exh's affair and wish I'd had it years ago; it helped me work through not just my marriage but my entire life and personality, or so it felt! And gave me a push in a direction I'm still following, without counselling.
You don't have to do all the logistical stuff instantly. We found an interim solution that gave us both the time to calm down and work out what we wanted in the longer run. And even if you just sort things out directly, honestly it breaks down into a load of smaller steps that just get done.
How do you think you might sort things out, roughly? Could you both have the kids, 50-50?