So I got this figured out today. It turned out to be a mistake made by the installers. Chances are none of you will run into the same issue, but I started the thread, so I thought I should post back and finish it.
I pulled the stereo out this morning. They had told me that they spliced into the accessory outlet wiring for power, because the accessory wire in my truck's radio wiring harness was always hot. I checked that with a meter, and found that they were wrong, as I suspected. It went hot and cold with the ignition. I disconnected their power wire and wired it up properly to the truck's harness.
I started up the car, the stereo came on, then I shut off the car, and the stereo stayed on, as well as my trip computer, temperature reading, odometer, and clock. I powered the stereo off with the power button, and it turned everything off. I pulled everything out again, and cut their zip ties so I could check the wiring between the two wiring harnesses. They for some reason spliced the power antenna wire into the 12v accessory wire. They did this on the stereo's wiring harness, so it wasn't like they just mixed up the wires when they were making connections between the two harnesses.
I can't think of any reason why they would have done this. With it disconnected, all of my issues went away. I'll be calling them on Monday to see about a partial refund, although I don't expect to get very far with that.