Real Estate Law
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Well, I don't belive I would sign the document as it doesn't put you in any better position than if they foreclosed except maybe creditwise. If they can still sue you for any deficiency, then it is not a true DIL transaction as a DIL typically just cancels the mortgage and note and the bank just gets the house back.
I would definitely cross out the statement, initial it, and then send it back to see if they would agree to go through with it.
In order to claim NC as your domicile for residency requirements, you typically have to live there a year and intend to make NC your home permanently. So if you claim it and they dispute it and show that you only did it for the exemption, then they could claim fraud on a creditor and reinstate the deficiency balance.
So the long and short of it is that it is unlikely that they would pursue you for the deficiency as they would have to go to the time and trouble to prove it was a second or investment home. But I wouldn't want to risk it without trying first to mark out the clause and see if they would agree. I would feign ignorance if they questioned it and explain that you thought that meant that you would automatically have to pay any deficiency.