To my understanding, there are some lenders out there that will indeed do what you've suggested ~ take a note for the balance. But it depends a LOT on the situation, your financial condition, the amount of the note, etc. You see, the lender will be giving up their collateral (the house), so the "loan" would basically be an unsecured note. You really should discuss that idea with the lender that holds your mortgage.
All the best!
The other alternative is Loan Modification. Have the bank alter the rate to give you an affordable payment. It is time consuming, and not guaranteed but has a happy ending if the bank agrees.
I have seen short sales, take a difference and swap it to some form of credit line. usually they put it on another piece of collateral. But I have only seen it happen on a 2nd note. not on a first mortgage. But you could try it. It doesn't make sense to me personally, though.