Unfortunately, the way underinsured motorist coverage works in AZ, a 15/30 policy will not provide any additional coverage where the at fault driver has 15/30 liability coverage. This is because UM coverage only applies to the amount in excess of the at-fault driver's liability coverage up to the insured's UM policy limit. For example, if you had 50k in UM coverage and the at-fault driver had 15k in liability, you would have an additional 35k in UM coverage benefits available to you. But where the at fault driver has a 15/30 liability policy (the minimum required by law in AZ), a 15/30 UM policy would not provide any additional coverage.
So, it would seem that an individual in your circumstance is essentially limited to what you can recover from the at-fault driver's insurance or the at fault driver's personal assets. Here's the difficult situation you face: in order to go after the at-fault driver's assets, you need to file a lawsuit and obtain a judgment. That takes a lot of time and money, and the insurance company will not release the policy limit until you've obtained a judgment because that is their incentive to prevent you from suing. So, to have even a shot at collecting from the at-fault driver's insurance, you need to go to trial and incur those costs, wait a year or more for your day in court, and obtain a judgment. The insurance company will pay the policy limit at that point in time (assuming the judgment exceeds the policy limit) and you would then be tasked with pursuing the remainder of your judgment from the personal assets of the defendant.
The thing is, most individuals do not have the money to pay a large judgment. So, you will likely then need to engage a collections attorney to obtain a wage garnishment order, put liens on real property, and see if there are any liquid assets that can be seized. This again costs time and money. And at ANY point in this process, the at-fault driver can file bankruptcy to discharge the debt, leaving you with no further recourse except as a creditor in the BK proceeding.
All of this is to say that while pursuing a judgment and attempting to collect from the personal assets of the at fault driver is certainly an option, it is usually not a practical one. Generally, it is only advisable if the at-fault driver has substantial assets (i.e. they own a valuable home, business, are well known in the community, etc.), as in those cases they will likely be able to pay the excess judgment or at the very least are not a huge risk for bankruptcy.
I realize this puts you in a difficult position, but this is the legal reality of your situation and you may be forced to choose between the "less bad" of two options as there is a very real possibility that you'll have no way to ever receive full and complete compensation for your damages.
I hope that you find this information helpful. Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.
If you do not require any further assistance, please be so kind as to provide a positive rating of my service so that I may receive credit for assisting you. Very best wishes moving forward.