The liable party in my case offered to pay me not to file a claim with their insurance. Should I accept?

No. You should never accept cash in lieu of filing an insurance claim. You are at risk of not receiving the cash as promised and can run into difficulty if the amount you are given is not enough to cover your expenses. Instead, you should obtain the other driver’s name, contact information, driver’s license number, the name of their insurance carrier, their insurance policy number, and the vehicle’s registration information. Then, you should file your claim legally.