A common solution that the courts will often agree to, is for you to remain in the house with the kids until they are 18, YOU make the mortgage payment (think of it as a kind of "rent"), and when the youngest is 18, the house is sold and the profit is split 50:50 between you and ex.
Of course, you would be better off if you could buy him out of the house now, and keep that appreciation for yourself.
You need to speak to your attorney about your options.
Bear in mind, though, also, that if you try to keep the house, houses can be expensive when it comes to unexpected repairs, especially if your house is more than 10 years old. Don't try to keep the house if you will not be able to afford to fix roof leaks, buy new water heaters etc.
And definitely don't agree to a sale to go through until the divorce is final. This is your big bargaining chip, the courts will not force you to sell before ALL of your divorce agreement is in place.
My ex pushed me to sell our house right after he left (because he wanted to buy the duplex he was renting). I did go along with it because I knew I couldn't afford to buy him out of the house, but in retrospect I should have held out for a while - it put me in a bit of a precarious position, as we didn't finish our divorce until 1 1/2 years after the house sold, and while we sold at the bottom of the market, I would have been out of luck if the market had started to take off, since I couldn't buy a new house until the divorce was final. (I'm self-employed, and make enough to qualify on my own, but with the new more stringent lending rules after the crash, they didn't want to lend to me until the divorce was final and I had a year's worth of cancelled alimony checks!).
You need good advice from an attorney, don't agree to anything without legal advice.
