I have no idea on how you do this but I've heard of people subleasing. You may be able to get a place for a couple of months that way. It may even be a house. The only problem there would be no furniture. But if you don't mind moving in your own furniture it may work. You can live on air mattresses for a while, too.
There may also be people who need to go away themselves for business trips and they may sublease with a furnished place.
I think your best option may be to talk to a real estate agent in the new area. They manage properties and may be they can make an exception to the regular rules about leasing. You'll need an agent when you buy your own place; so, they may work with you for the future sale of you buying another place. I presume you will buy when your old place sells. There may be places you can get with the option to buy, but rent now.
Maybe, take a new look at different options like renting a campsite to live for a while. Maybe, you can rent an RV that would be cheaper than rent. I know you can get those for any time spans you want. Going cheap like buying a tent could work. Fortunately, summer is coming; so, this is doable.
If your family is willing to take on one boy then maybe think about splitting your family up for a few months. If everyone lives close by it may work. You keep the one that you feel needs the most help. Offer to pay other family and maybe everyone will be more open to the idea.
Sometimes, it could be cheaper to get two resort rooms with a door between rooms at a cheaper hotel. Some are set up; so, the kids don't have access to a door in both rooms. These places will give you a better deal if you're doing it for a month or longer.
Finally, there's no real reason you need to move away to your new area. Think about renting where you still are; it maybe a cheaper option.
I'm just trying to think out of the box; you never know what may work.
I have a friend selling their place now and it's looking very dismal. Their place is absolutely immmaculate, updated, and in a good neighborhood. They've been on the market for a month without even a look and they have a chain realtor. They're in a suburb away from a big city. They haven't had an open house yet, but it looks like they will already have to reduce the price. She is thinking it's going to take at least 6 months to sell in this market. They are prepared to take a beating. I'm telling this story because you may be very optimistic about selling your place in such a short time.