After a North Carolina car accident, do I have to get my car fixed before I collect for damage to the car?

No, it is your car which has been damaged and you and only you get to decide whether to fix your car.

The negligent party who caused the wreck or their liability insurance company is responsible for paying you the damage to your car even if you choose not to have it repaired.  

The amount you are entitled to receive for damage to your car is the difference between the value of the car immediately before the collision and its value immediately after the collision. The repair bill or the cost to repair your car should you choose to repair it, is only one of the factors in deciding the before and after value. You are also entitled to an amount to account for the fact that your car is now a wrecked car. Consequently, you cannot sell your car for as much as you could have before the wreck even assuming that your car is fixed properly.