Can you just say to him what you've told us LNM? That your DDs are asking when they can see him and that it's now his responsibility to make a cast iron arrangement to do so? Maybe reaffirm that this isn't about you / your marriage, it's about his children and that sometime soon there (may) will be a court order in place regarding access so he needs to get used to making them a regular and non-negotiable part of his new life.
Forgive me if this sounds a bit self-help-book-y (I'm a personal coach among other things, underneath this normal exterior!) but regarding work can you think about what your ideal outcome would be? Maybe it's a new job, maybe it's your boss knowing about the situation but not having to talk to him/her about it, whatever... Then you can take the steps needed to make it happen. If it helps imagine one of your DDs in your shoes and think what you'd advise her to do. Could you email your boss the bare bones of the situation so that they know about it but you don't have to discuss it face to face? I there a friend you can ask to help with brushing up your CV and start looking for something else? Anything you can do to be clear about what you want, so that you can take small manageable steps towards it, will make you feel better.
I find that when I have a situation I dread, the best thing to do is look at worst case scenario(s), plan for how I'd deal with that / them and then anything else is a bonus. Plan for the worst, aim for the best, and be positive. Ask for the help you need. And have some
I can't imagine how draining all this must be xx