I’m sorry you are going this OP, it sounds tough for all involved. Relationship break down is more common with couples who have children with additional needs, and I very much understand that life must be hard work.
You need support with the children and the long and short of it is that you’ve kicked the other carer out. So I guess it’s a case of building a new support system, which you will do time.
I know you say technically it wasn’t an affair but I do feel this is a moot point. Affair or not, it doesn’t change the fact that being pushed away like that must really, really sting. The technicality of whether it was or it wasn’t cheating doesn’t matter: it doesn’t make you in the right and your ex automatically in the wrong. Playing the blame game won’t be helpful.
So now he’s gone, you need childcare and want him to step up at the times that suit you but he won’t. I can actually see why he is not being very helpful, depending on how forthright and pissed off (like you sound on here) you are being with him over this.
You might find that taking the emphasis off you and your needs, and shifting the focus solely on the children, co-parenting and their relationship with their dad as a more beneficial strategy. He’s their dad not your babysitter.
This is a huge adjustment for you all. Mediation might be beneficial. Try and be kind to one another - you did say he’s a good guy, so hopefully in time it will all improve and you’ll get into a new routine. I know plenty of couples who have split up very acrimoniously but eventually found their new groove and now work well together, even with extra complications with children’s needs and new partners. It can be done.
Good luck.