It's hard to move out of your home when you divorce. But, most of the time coping with the move is much worse for the adults than it is for the kids, particularly if the kids stay in the same school.
You've gotten lots of good advice here. Most divorce attorneys will recommend selling the jointly owned home and splitting the proceeds at the time of the sale. It's just the cleanest method to get things financially separated, which is good for everyone in the long run, and also they've seen plenty of single mothers try to stay in houses they can't afford to upkeep.
Yes, your sister needs to have a nice place to live. I remember my dad telling me that when I was divorcing, sad and upset at having to move out of my nice house, and worried that temporary finances would have me living in the ghetto. So, thank you, Papa Zamboni, for suggesting that I check into the rent at the "luxury" apartment complex closest to my children's school. It cost more than I thought I could afford in rent, but still way less than the house. That property had a heated pool, a "waterfall", ponds full of frogs and crayfish, an indoor basketball half court, a gym, and very high density of their school friends . . . the kids LOVED it. They were WAY WAY happier sharing a bedroom with mattresses on the floor in my tiny apartment than they had been in our hoity toity suburban house. In retrospect, I'm so glad I got out of that house. Besides the kids being happier at the apartment, it was too big and a money pit even though it was fairly new (big yard, big utilities bills, etc.)
Like your sister, I couldn't afford it. If I had stayed in that house with "help" from a relative, then I wouldn't have nurtured the new sense of independence that I needed to get my life straight. Also, about 15 months after we separated, my ex stopped paying child support for no good reason. He just decided he resented writing checks to me and stopped, rationalizing in his own mind that he paid plenty for the kids' expenses directly already. I would never have predicted he would do that, so you just never know. We had a signed agreement about what he would pay, and social services helped me recover that money, but it took a good six months of not being paid before they even got it on their docket. Thankfully by then I was out of the house and in a situation that I could pay my bills every month without that money.
Good luck with it! My suggestion is that you offer to help your sister move if you are nearby.