Everything purchased or obtained during the marriage is marital property. With a house, you can sue for the difference in value between when you were married, and the date of your separation/divorce. So, if the house was worth $100,000 when you wed, and $120,000 in the end, the difference is $20,000, of which you would be entitled to half. Be aware that other factors, such as retirement plans and assets and liabilities, make this number subject to augmentation or depletion. It’s called Equitable Distribution for a reason.
You both have to decide on who will leave the house–but as a man, I would want someone to shoot me in the head if I EVER asked my ex-wife to take our kids and leave. As much as I didn’t want to be with her, I would NEVER sit in the house while my children had to adjust to something that was ultimately my doing. How horrible. Maybe, since it is his house, you could stay there and pay the mortgage while he gets an apartment–and then agree that at some point, you will find your own place to live…one without so many memories (or ghosts).