what ifs

What is the name of your state? South Dakota

Situation: I have only liability insurance on my car.

Next Situation: If someone hits my car and they are at fault.

Question: Can I sue striaght up for the damages to my car, or will my insurance try to get the claim from thier insurance company?

you will need to sue the other party or make a claim on their insurance policy. your insurance will not make a claim on your behalf for property damage.