Well, that was a bust. Got about 30-40 minutes of runtime and right back to the problem, and a continuation of the problems with the power button. I removed the heatsink, cleaned off the thermal paste from both it and the CPU, but now I can't seem to find any of the 3-4 spare containers of thermal paste I had sitting around. Always the problem, right? Keep spares of something that are always in your way until you need them, then you can't find them.
Given the exact same power issues I was having before swapping the PSU, though, I'm starting to think that maybe the mainboard is faulty. I've never had a machine either reboot, or not turn off, or some weird combo of powered states after holding down the power button. They always just shut off. After my second attempt at running the machine following the PSU swap, I couldn't get it to actually turn off until I manually disconnected the power. I encountered similar issues before the swap, but like I said before, I was just dismissing it as power button issues. I don't think that's the issue, anymore.