Yeah, whoever built this thing did a ham-fisted job of soldering it together. Appropriate, because I found it at a ham radio-related flea market...
I'll have to try cleaning the old flux off and check the joints when I go in there to re-seat the ICs. I'm not entirely convinced that the issues with the board are entirely due to the builder; my friend (who's an engineer) was cursing the size of the holes in the board, as well as the fragile condition of the wires connecting the switches and such to the board. I'll take some pictures of the component side of the board when I next remove it from its stand-offs.
The most annoying part of this clock's issues is that it seems entirely random; it'll run hours without a glitch, or it'll glitch up within a minute or so of being plugged in, and then run glitch-free for several hours. Also, there's no consistency to what it does during the glitch; it'll skip ahead several hours, or just a couple of minutes. In addition, tapping on the casing doesn't have any noticeable effect on how the clock works.
All in all, it definitely looks like I inherited someone else's problem.

I can't help but wonder if this glitching issue is what a previous repairman was trying to fix by replacing most of the ICs. As they say, you get what you pay for, and considering that I paid only $5 for this thing...

-Adam