Well, the criminal side and your civil side are completely unrelated.
Your local authorities and the district attorney's office will handle the criminal act of leaving the scene of the accident, if you press charges.
I'm not a believer in running to a lawyer for every single thing that happens. Definitely go to a doctor to get a complete once-over to make sure you're OK, even if you only seem to have minor surface injuries. Then see what offer you get as a settlement. If it covers the fair value of your car, the doctor's visit(s) (assuming you are not in a no-fault state, where everyone covers their own medical expenses), the time lost from work, any additional expenses you incurred, does not close the case from additional medical claims (again if not in a no-fault state), and does not contain a gag order... then take it and walk away that you were compensated.
I hate to say it, but you are only entitled to the value of the car, not the payoff.
However, if they start screwing with you in any way, or will not agree to any above item, then get a lawyer involved.
