I know this is an old thread and you may be long gone but anyway, here goes.

I am in the process of dealing with the same issue on a KEMC308 unit stays on. Keypad and display indicate unit is off and functioning normally dispite it heating on the inside. Fortunately for me I have a spare KEMC378 unit stored in my garage.

First I swapped the control panel and expected joy, no dice same problem.

Next I tore into the heating elements. The round element around the fan was the offending part. It was a little cruddy around the end where the plug connects, best I can figure is it was shorting to ground and feeding itself enough voltage to get and stay hot. I swapped that element with the one in the garage and BINGO back in action, good thing as tomorrow is Thanksgiving and I have got family coming over!

If you are still paying attention to this thread cbo2010, and you still have your old oven it may be worth your time to check the elements, and maybe spread some joy to the professionals that have already gave it their best.

Best Regards,
t_bare

Straight dope message board, huns