"We're about to get a divorce… so we're buying a bigger house together. Well, I say 'together', I'm paying for it and it's in her name."
Wait. Wut? Will you still be lodging there (and paying for it) when her lover moves in too?
Point the first, been there, "woman I love" etc etc. It's singularly difficult to extricate yourself for all manner of weird and wonderful reasons, and you'll be racked with guilt and pain and all the rest of it. But, get out you must. When you do, in five years time I promise you, I absolutely guarantee you that you'll be sitting there thinking "dear gods, how could I have put up with it for so long, why didn't I do that years ago?"
Point the second, c'mon, read back. Read your own posts. If a mate came to you and asked you the same question, what would you tell him?
She doesn't love you, she realises she's stuck with you hanging round like a bad smell, and has taken the decision to bleed you for everything she can. If you go through with this arrangement and then decide stay with her, hey, free house! If you decide to MTFU and leave, you're out on your ear with precisely **** all. Whereas now, you hold all the cards. She cheated, all the property is in your name, she has nothing and no leg to stand on, and you're about to hand it all over to her. Why? Love? Guilt? Because she's told you to?
By all means, be a good father, support your kids, but you can do that without having to get such a sore bottom in the process.
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