Replaced the control board. Total process took me about 25 minutes (I guess I move a bit slower than most of the folks here).
The new board was a perfect fit, and actually seems to be an "upgraded" version in that the display screen has various animations that the old control panel didn't.
Since I've now fixed both the washer and dryer myself in this paired set, I figure I've saved about $2000 on a new matched set. Totally worth it.