Re: Insurance company does inadequate investigation
You sue the other driver, and the owner of the vehicle (because the owner is liable by statute for the negligent acts of the driver). You don't have to wait for anything.
When you sue them, their insurance companies will tender a defense on their behalf, and the insurance coverage will be in play (unless they are disclaiming coverage based on some exception).
But more fundamental is that personal injuries sustained in a car accident must satisfy a certain threshold before you can sue. NY's no fault laws only allow you to sue if you have suffered "serious injury". You need to consult a lawyer to find out what that means, because it is the source of much debate and litigation. Certain injuries clearly meet the threshold, but most are in a gray area or insufficient.
If your injury does not meet the threshold, then you are relegated to collect no-fault benefits from your own insurance carrier.
Listen - insurance companies constantly do this. They don't want to pay you. They will try and nickel and dime you and draw the line and dare you to cross it. So assess your case with a lawyer. Most personal injury cases are accepted on a contingency basis anyway. Many times you need to file a lawsuit to get satisfactory results. You have a 3 year statute of limitations, so be mindful of that, but also be mindful that once you start a lawsuit, it may not settle or be resolved for another 1-2 years, so better to sue now rather than later.
Feel free to contact me for a consultation if you wish to discuss it further.