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Is house swapping possible?
Another possibility that you might consider is to find a renter for your current place and move to your destination. If you have a lease for your renter, your mortgage company can consider that as income for approving the purchase of a new house. I did something similar when I purchased my current home, but I was also able to get approved without selling or renting the old place. There's no reason that someone couldn't create a house swapping site for longer-term than a week. It may not initially have as much demand as a 1 week swap, but there are no such existing services that I am aware of.