An uninsured individual hit us and was at fault, can I sue for the damages?

If the accident wasn't your fault, your premiums shouldn't go up because of it;  complain to the state government if they try it and don't back down when you complain.  And if your insurance is paying, you can be quite sure that they have the police report, and intend to do what they think makes sense to recover the damages if they can.

Anything you'd win for property damage to your car would go back to the insurance company, aside from anything you had spent on the repairs yourself.  It's also possible that your insurance company will sue in your name, under a principle called "subrogation," and if you're asked to cooperate with that, you must.