If I owe more than the amount they offer can I negotiate or refuse the amount they offer to pay?
No, If you refuse, you will get nothing. You can try to negotiate a better price though. What you will get is called Actual cash value (or fair market vale) and is based on what other like cars in like condition are selling for in your area. What you may owe on the car has absolutely nothing at all to do with anything and will not be considered in whatever you end up with for the car