If you let a friend drive your car and he runs off the road and does major damage to your car will your full coverage insurance cover it?

It depends on the policy. Your policy may have something in the fine print about allowing another person to drive your car. When you got the policy, you agreed to be responsible which means you are the only one driving the car. You can file a claim with your insurance company, but you will pay through the nose with higher premiums and they may drop your coverage altogether.

Since it was your friend who was driving, he is responsible for the damage. I saw a case in court where the friend was successfully sued for wrecking the owners car. Ask him to pay without a fuss or take him to court. Don't involve your insurance if you can possibly avoid it. And don't allow someone else to EVER drive your car.