After cleaning the dust out of the Alphaserver, and taking a pencil eraser to the edge contacts on the old CPU board (the 500 MHz one hasn't arrived yet), the system appears to be running solidly. I hope it'll continue. I'm not going to declare it fixed until it's run a week straight without problems, though.
The system smells like cigarette smoke. I suspect that the crud in the air got onto the contacts and screwed them up. Let that be a lesson to you out there: Don't smoke, especially around your computers.