Finalize the sale of the car I listed.
The buyer will be paying cash when they receive their check this coming Friday
from their insurance company. How do I proceed to from here? Should I pay
off the rest of my loan?
The buyer would be a fool to buy your car if it has a loan on it unless they know the loan will be paid off in Full. You still owe the money even if you don't have the car anymore!
You'll receive payment for a car for which you do not have the original title. If you explained all this to the buyer, he's probably too stupid to drive. If you're honest and have cash to pay off the lien, do so. Or take the buyer to your lender's office and have him process the sale there. All financial obligations will be dealt with and the buyer will be mailed the original title from your lienholder.
Also have a look at this link: https://cargurus.helpscoutdocs.com/category/273-p2p-transactions