Hi Olivander
You and your father in law should be well aware of the risks involved in this transaction. You buy the new house and your "easy to sell" house doesn't sell because the market starts falling. You are left with two houses and a very expensive bridging loan.
If you are both prepared to take that risk that is fine.
But you should put your house on the market immediately. If it's easy to sell, then you can put in an early closing date. If you have a signed contract returned to you, then you can sign the contract for purchase.
If the closing date on your sale is after the closing date on the purchase, maybe your FIL could borrow the money himself and lend it to you.
Could your FIL buy your house from you? It's messy but it's a solution.
But it's very risky buying another house without having exchanged contracts on the sale of your own.
Brendan