I'm sorry that you've had to go through this. Car accidents are rough business (having been through two, I know how bad they can be!).
As others in this thread have pointed out, Utah is a no-fault state. Legislators intended for this to cut down on court costs and unburden the legal system. Drivers just turn to their insurance companies and have their damages paid for by their own carriers in what is known as a "first-party claim." This insurance is supposed to pay at least $3,000 in PIP (personal injury protection) benefits, no matter who is determined to be at fault in the accident.
In theory, it seems like a great idea! In practice, many drivers end up without anyone to pay their bills because their insurance companies deny the claim. It also ends up being pretty cost-heavy on the drivers. Furthermore, if a person is going to file a lawsuit, that individual must have exhausted the PIP benefits or must have crossed an "injury threshold." That person has to be grievously injured. We're talking dismemberment, permanent disability, amputation, disfigurement, etc.
So in this case, you can't file suit unless your injuries are grievous OR your medical bills are more than the $3K threshold. Your insurance company should be paying them otherwise. Then your insurance company would be reimbursed by the at-fault party.
There is technically fault here, even if we're talking about a no-fault state. No-fault insurance has limits. It pays benefits for medical bills and lost income only -- it doesn't take care of your car, pain or suffering, or anything else. Those are separate claims. This is why claims and lawsuits above and beyond PIP are allowed, since $3K doesn't go very far in terms of what needs to be paid following a car accident.
Now, if you think your insurance company is unreasonably denying your claim, that's a different story entirely. They "couldn't contact the other party"? In today's age, that seems odd. Did they only try once? Did they use smoke signals?
Additionally, Utah is a modified comparative negligence state that follows the 50% rule, which means that if a driver is more than 50% responsible for an accident, that driver is barred from recovering any damages. It sounds like your insurance adjuster didn't quite know what to tell you and was just saying whatever came to mind.
In terms of them denying your claim, if you think they were acting in bad faith, first try speaking to a supervisor. Tell that person you think your claim was unreasonably denied, but make sure you have your ducks in order. Have all of your documents together and make sure to write down every name of every person you speak with, along with the dates and times. Be polite, but don't back down.
Clearly state that you were zero percent at fault (or whatever percentage it actually was, as long as it's less than 50% -- considering you were T-boned, I can't imagine it's more than 50%). As such, you need to have your medical bills covered. Now, PIP won't cover your car, but I don't see why your insurer wouldn't cover an accident that you clearly couldn't avoid.
If your insurance company still won't relent and you feel that the denial was unreasonable, contact your state insurance authority and ask them to investigate. This will put some fire under them, and maybe they'll get penalized for their actions. Additionally, you can think about hiring a personal injury attorney to help you collect from your insurance company. Make sure to keep a file or Excel spreadsheet of all bills you've paid to date so he or she can try to get them reimbursed for you.