Well, today I was able to play for a few hours this morning, and then it started shutting down again, even with the lowered resolution/graphics.

cosmicarmagedon - It's a pre-built (from Main Gear), running AIO, and it usually sits on a piece of finished wood a little bigger than the footprint (the case has "feet" to allow air flow so I definitely keep it off the carpet). I went into the BIOS and adjusted the CPU fan speed to Turbo, but unfortunately it didn't seem to help - after rebooting, I logged in, and it shut down just as I was about to load into the game.

On the (sort of) bright side, I was finally able to replicate the issue with a game other than FFXIV. Playing the Witcher 3 at max settings, it shut down about a half hour in, as I was about to enter a cutscene. Now that it's not just happening with FFXIV, Main Gear agreed it must be a hardware issue, and I'm going to be sending it back to them so they can start replacing things and troubleshooting, until they (hopefully) find the issue and fix it.

Thanks to everyone who replied with suggestions!