If or when it is clear that the insurance company is unwilling to cooperate despite clear evidence of their insured's negligence, your attorney will file a lawsuit against the defendant.
It's very important to remember that your attorney would be suing the person that harmed you - not the insurance company. The insurance company will try to settle on behalf of their insured or they will hire lawyers to defend their insured in a lawsuit.
It's true! Even though it is fairly common knowledge that motorists are required to carry liability insurance, we are not allowed to mention it. This is so that the jury is not swayed into thinking the defendants can or cannot pay the potential amount the jury might award to the plaintiff.
This is a result of the power that the insurance lobby has over Texas law. That's a big win for the insurance companies but not always for the injured party. Insurance can prey on the sympathy the jury might feel for the defendant's ability or inability to pay for the plaintiff's damages.
On top of that, we have to pretend as if the defendant will be paying out of his or her own pocket for any potential verdict even though it is the insurance company who will pay.