First and foremost, you need to be aware of the statute of limitations. A statute of limitations is a time bar within which a claim for personal injury needs to be brought. For example, I practice law in Oregon and the statute of limitations for personal injury in Oregon is 2 years. Therefore, a personal injury claim in Oregon must be settled or a law suit must be filed within 2 years of the date the injury occured. Otherwise, the claim will forever be barred. You should check with an attorney in your state immediately to determine the statute of limitations.
Before settling your injury claim, it is very important that you first complete your medical treatment and become "medically stationary". "Medially stationary" means that your injuries are completely resolved or have otherwise reached a point where further medical treatment will not substantially benefit your condition. You will want to be medically stationary before settling for multiple reasons. First, as part of any settlement you will be required to release the other party and their insurance company from any further liability. Obviously, you do not want to prematurely release the at-fault party and then incur additional medical bills that you will be stuck with. Second, you are not only entitled to have your medical bills paid, but you may also be entitled to pain and suffering damages. Of course, you don't know what your total pain and suffering is until you have fully experienced the full course of your injuries.
As far as paying for your medical care now, it sounds as though the at-fault's driver's insurance company is taking care of medical bills for you. You may also want to check with your own auto insurance carrier as you may have "personal injury protection" or medical coverage that will cover your injuries in the event of an accident. Generally, with this type of coverage, your own insurance company will pay for your medical bills and then seek reimbursement from the at-fault party or their insurance company. Finally, you may also use any personal or private insurance that you have, either through your work or that you contracted for privately with an insurance company.
If your doctor is unwilling to accept the insurance coverage you have available to you then it may be time to find a new doctor.
Sign up to receive a 3-part series of useful information and advice about personal injury law.