Does insurance cover car damages while driving out of state?

Yes. Your insurance adjusts to the higher or your liability limits or the minimum limits for the state that you drive into. Be aware that all U.S. insurance will cover you in Canada as well but not in Mexico. When you cross the border into Mexico your insurance does not. If you move to another state you are required to notify your insurance company and if you don't it could be a cause for denying your coverage.