What you really need is a wiring diagram so you can figure out how the timer motor is getting power when it should be off.
I guess you did not find it in the control console.

Your unit is actually a Whirlpool and I cannot find any tech info on it and do not know of a cross reference for Sears to Whirlpool units.
So the below is a lot of guessing on my part.
I do not have access to Sears wiring diagrams though you may be able to purchase one from them.
I think is is a Sears 70 series.
Here is an example wiring diagram for a Whirlpool unit, with luck it will be close to yours, see the attachment.

Did this unit ever work correctly for you at this location as it sounds like it is a wiring problem.
A couple things jump out at me:
1. The timer should not be able to run with the door open in timed dry as the switch disconnects Neutral from the unit and it is required to run the timer.
2. The heater should not be able to come on when the motor is off as the centrifugal switch on the motor should disconnect the heater.

Perhaps if you try it in an auto cycle it may give you clues to the problem.
In auto the timer motor gets power through the heating coil when the coil is shut off by a thermostat. The resistor then reduces the voltage from the 240 down to 120 for the timer motor.

The only thing I can think of and it does not make total sense to me is that you have a grounded element and it has grounded is a way that is doing weird stuff.
Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case.