Both of you will have court fees. Whoever files first (Petitioner) will pay a filing fee and the other (Respondent) will pay a first appearance fee.
It doesn't make any difference who files - it does not put you in a better light to have beaten the other one to the punch.
If neither one of you want the house, why not get it ready and put it on the market now with the understanding that you each get 1/2 of the proceeds. The court will enforce that even if you sell before the final property order is entered by the court.
Your local courthouse may have what's called a Family Law Advisor who can help you.
If I were you, I would go to the bank with hubby and close out all accounts with 1/2 of each account going to you and 1/2 going to him. Then, I would open my own account(s) and start looking for a place to move to. Once you've done just that much, you will feel better because you won't have to dread going home every day.
Also, go ahead and get the paperwork going for the divorce. Doesn't sound like it's going to be contested. If you and hubby are able to, work out a custody and visitation plan and division of the assets.
If you guys can work all of this out, the divorce should go a lot smoother.