Thanks to all of you for your help! The key was to clean out the plugged vent. I didn't know that it existed, much less that it
could and should be cleaned periodically. I popped that cover off and it was GROSS in there - years of sludge. It looked about 80% blocked. I never would have guessed that improper air-flow would cause it to splash over the bottom dam.
I was lucky to find a helpful community of dedicated Appliantologists.