I wish there was someone who could come and mother you, because it's what you need right now. You said your Mum changed the subject, would she come if you said you needed her, or would she be more of a hindrance? Is there anyone who could come?
At the moment, this feels like too big a mess to you and you can't see the wood for the trees.
The sex you're having is understandable, but in this case it's doing you no good, because you absolutely need to detach from him. For you, it's also loaded with completely the wrong agenda. You're competing with the OW, but the reason he won't have penetrative sex is because he considers that would be infidelity to her (and I know how fucked-up that logic is, believe me).
He's duped you for 5 months and abused your generosity in saying you would try to forgive. When he got found out, he appears to have told you that he will be leaving you, but only when it suits him/them.
Don't let this happen. You need to get away from the sight of him, his smell, the sex - and all the things that keep you attached and addicted to him. You also need to restore your own self-respect and believe me, whatever happens, you will eventually want him to respect you too. He doesn't right now.
Tell work what has happened and ask them to cut you a bit of slack for a short while. Find out if there's a workplace counselling service too. See if a RL helper will help you sort out finances, because the more things that you can get some clarity on, the better you'll feel.
Don't worry at all about giving yourself and the DCs an easy life at the moment - no child ever died from tinned food and ready-meals. Ignore the dust too. Treat this situation as though you have been knocked over by a bus and need help to nurse your wounds. But the most important thing you can do is to muster up all your dignity and ask him to go now.