If a lender notifies you that your loan is paid in full but you actually owe $7,000, do you still have to pay? 

If they come after you and they have documentation showing that $7k is still owed--and you in turn do not have cancelling checks or the like proving payment of it--then they can force you to pay; the fact that someone in their organization made a paperwork error does not negate your obligation to pay. However, having that letter does mean that you could claim in good faith that you thought it was settled; if you do not continue paying and they notice their error and come after you, you would have a very good defense to late charges, penalties, etc. and would most likely just need to repay the remaining balance.