Slow down and think this through. I can't come up with any scenario where it makes sense to buy a new house with this woman at this time, even if she's not on the deed and even if you can afford it on your own. Not being on the deed doesn't necessarily mean she can't claim half of the equity value on her way out the door. And could you still afford it if she took half of everything else that you own (think half the furniture, half your cash savings, (possibly) half your retirement, and on and on; what if you have to pay child support and alimony for some number of years? Would the house be a good one for you if you are in it all by yourself at least half the time, or would it be a lonely place until your D14 came for her visitation?
I think you're hoping that her apparent interest in buying a house indicates she wants to reconcile and go the distance in your marriage. It does not mean that, unfortunately.
I guess it isn't specifically stated in Sandi's Rules that you shouldn't buy a new house with a wayward spouse, but I kind of think maybe she thought that was so obvious it wouldn't need to be said.
I was a big pursuit guy, too, so don't think I'm judging you. Just hoping you see the light before having to go down a bunch more painful, cheese-less tunnels (like I did).
Me: 46 W: 44 Married: 17 Together 21 D13; S10 BD: 03.03.15 (Not attracted to you) Almost 2 years trying, alone, to save marriage Status now: Divorced (effective 06.13.17)