Hooray! I believe I've found the source of my problems. First off, there *was* a short on the board:
Notice the tin "whiskers" connecting the center pins to the pins directly to their right. Cleared those out, and I thought everything would be fine....
But it wasn't. So I pored over the whitepapers, ran Ohm's law calculations till my head started spinning, and finally went to bed really frustrated. But then I had an idea -- what if my problem isn't with the design at all? What if I was just over-driving my power supply?
My power supply is a cheapo Micronta 13.8V, 1.75A unregulated supply that I got at radio shack a zillion years ago. Could *that* be the problem?
So as an experiment, I wired up an Antec compute (regulated) power supply to the system. Bonus: computer power supplies have big fat 12v *and* 5v rails (24A on the 5v rails for my 500W supply!!). And wouldn't you know -- everything worked fine. The clock didn't get all wonky, and the program counter ticked just fine.
So next step: rip out all of my 7805s from the design and leave power generation and regulation to the power supply. I'm just going to solder in a standard Molex 4-pin power supply header to the control board, and distribute the 5V and 12V supplies like usual.