What happens when you get into an auto accident that is your fault and the demands of the other drivers claim is higher than your liability limit on your insurance?

The other driver has a right to sue you to recover any damages not covered by the insurance. It could mean a lien against you and against any property you own. If you have nothing thay can take, let them sue. They can't get blood out of a stone and they can't take what you don't own.