Since it was her that decided it was over, then it is she that should leave. Don’t give her the upper hand. You can’t be expected to suck up the pain as a result of her choices. She is worried about leaving the kids. That is her choice. It is a package deal. I suspect the balance of their time between you and her won’t be affected by that anyway.
If selling the house balances the equation, then by all means do that. Your kids preference if keeping the house or not is not relevant. They will adjust to whatever home(s) they are in.
I would start by documenting her actions (including the affair) and what she said and continues to say. It may be of value when it comes to how your imminent divorce and custody issues go. Also get some legal counsel as soon as you can.
Best of luck!