So, we tested the terminals on the LF surface unit with a multimeter (stove was unplugged), and it was getting power all the time, even when the switch was in the off position. As the lever was moved to a higher position, the multimeter would beep louder, and then softer as you'd slide it back to off. At one point, the beeping was slightly intermittent, but still very rapid as if the power was turning off and on. We tested one other burner that did indeed show no power when turned off, and vice versa, so something screwy is definitely going on with the automatic. We ended up disconnecting the power to the switch in the backsplash, so the LF is no longer getting power.
So, now we'll have to figure out how/why the automatic unit gets constant power. Do you think the thermostat is the culprit? BTW, all the wiring looks good (not even dusty!) and there is nothing obvious just from looks. I did take several pictures if anyone thinks those will help.