Why is my dishwasher drain pump constantly running?
Answer is in the comments to the OP, based on suggestions given there. 

Summary (but note that I may be calling some things by an incorrect name!):


Used a voltmeter to determine the voltage across the solenoid when I hit "cancel/drain" on the control panel. 120V. The solenoid/valve was thus most likely "normal closed".
Disconnected the leads to the solenoid and tested it with an applied 120V. Loud click/bang; the solenoid was clearly functional. 
Reconnected the lead wires to the solenoid, and used the voltmeter to determine that 120V was applied during the drain portion of the cycle, not the fill/wash portion, meaning that the controller unit seemed to be functioning properly. 
Removed the solenoid/valve unit from the frame/chassis and disconnected the tubes. Upon visual inspection, there was clearly a foreign object stuck inside that was preventing the valve from closing fully. 
Was able to remove the foreign object, put everything back together, and voila! It worked. 


[Note - I'm omitting all the other steps I took, when I was originally suspected that the controller unit was the problem. I still don't know what I did that originally caused that piece of glass to get dislodged from wherever it had been sitting for however many years, and get stuck in the drain valve. But the fact that I tried running a cycle that I had never used before seems to be ... unrelated. Or at least, I can't figure out the relation. It seems to be just a bizarre coincidence.]