If I was the injured party in an auto accident and was offered a settlement from the insurance company, will I still be able to sue the responsible party?

If you sign the release, that is the end of the case and you won't be able to sue.  By signing the release, you are giving up the claim against the other party in exchange for receiving the settlement check.  If you are dissatisfied with the settlement offer, you can reject it and continue negotiating with the insurance company to try to obtain a larger amount. Since you will need future treatment, the amount of the settlement should include the estimated cost of future treatment discounted to present value.  Once you settle the case, you won't be able to go back to the insurance company in the future and ask for an additional amount.

If you are dissatisfied with the settlement offer, reject the settlement offer and file a lawsuit for negligence against the at-fault party.  If the case is settled with the insurance company, NO lawsuit is filed.  If the case is NOT settled with the insurance company, you will need to file your lawsuit for negligence prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.

You mentioned the attorney's fee.  If you already have an attorney, and now go to a personal injury attorney, you will end up paying two attorneys.  If you don't have an attorney, it would be advisable to speak with a personal injury attorney.  The attorney could prepare the lawsuit and file it for you.  One factor to consider before filing a lawsuit in this type of case is whether or not you would receive significantly more from the lawsuit than the insurance company is presently offering.