What can I do if the at fault driver’s insurer refuses to pay for the full amount of damages?

An insurer will voluntarily pay what it believes to be the damages--or the amount of money to repair the car, or, if the car is worth less than the cost of repairs, to total it--if it thinks its driver is at fault. However, since there usually is some doubt as to fault (rarely is it 100% that one party is clearly at fault, and the other party did nothing wrong; there's also the issue of evidence and what could be proven), they usually discount or reduce what they will pay to reflect that  uncertainty. Also, since getting a settlement check saves the other party (the plaintiff, or one suing) the cost of a lawsuit, they will also typically discount what they pay to reflect that savings. Thus, the insurer voluntarily pays what it thinks the accident is worth, reduced by some amount for uncertainty and by some amount for not having to go to court.

That is not to say the insurer is necessarily right. And its determination is not a court adjudication. If they are not paying as much as you think you deserve, you have the right to sue the insured driver for the additional amount, so long as you did not sign a release or settlement agreement giving up your right to sue. (If you did, it is enforceable.) But bear in mind that winning is never guaranteed, and you will incur the cost of a lawsuit (e.g. a lawyer), and factor that into deciding whether to sue or not.