This mess is of her own making and now she's asking you to bail her out. That ain't fair, especially with OM in the picture and her making these choices.
I'm not vet but I'd say THE KIDS could stay with you and she's free to figure out what she's going to do on her own.
Failing that, she's right. If your name is still on the property, and the pipes break and ruin the house, the headache is partially yours; whether or not the responsibility to pay for the propane to keep the heat on is yours or not (and I don't think it should be).
ME: 38 BF: 40 T: 10y, no kids, no M (by choice) BD: 7/14/14, BF admits to PA, wants out, lies about new R. 10/1/14: I move out, BF lies about move in with OW 12/4/14: OW confronted, reveals all the lies