What to do if my family and I were in an auto accident but the responsible party's insurance won't cover damages?

You didn't provide much info so it's hard to give a good answer. I assume from your question that you believe the other driver was at fault for causing the accident.  Your best bet is to contact a good personal injury attorney in your area if anyone was injured.  Or you can try small claims court if no one was injured and the damages you seek are under the small claims threshhold.  You should also contact your own insurance company. Good luck.