Solution: I had a container (stopped) that didn't have a working icon URL. I deleted that container and things are good now. I wish unraid was better at handling these situations (or letting us disable pulling icons).
I bought a new PSU and new RAM and when I went to replace the PSU, I noticed that my one of my current RAM sticks was not all the way seated. I also noticed that the ATX 24-pin cable was a teeny-bit loose. Anyways, after plugging them in right, my current uptime has been almost 2 days with no hiccups.
If the server stays up for a few more days, I'll upgrade to a bigger case so my SATA power cables aren't pushing against the ATX port (which is what seems to be happening).
Consider this solved (I'll post if I notice issues again).