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Could be a grounded element.
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.
This can cause it to run all the time.

Note: You may have 2 problems as usually turning the timer to off should turn off the heating coil.
I could not find a wiring diagram for this unit so cannot confirm this.
The heater runs off of 240 volts. A grounded element is running part of the heater at 120 volts as it is now connected to Neutral. So you still have to have one side of the line connected to the heater. One side of the line goes through the centrifugal switch on the motor so when the motor is stopped that side should be disconnected. The other side of the heater goes through the thermostat's thermal cut-off and a set of timer contacts. So with the timer at off that side of the heater should be disconnected.

Since the only way to shut it off is to unplug it for some reason one side of the heater is still getting power which points to another problem.
I hope I explained the above OK so that it makes sense.

The above is a bit of a guess as I do not have the wiring diagram.
You may want to check in the control console to see if they stored it in there or it may be glued to the back cover.

[COLOR="DarkRed"]Could this be another clue to the heating element going out?[/COLOR]
Yes
What can happen is that the element grounds out in such a way that a short piece of it gets very hot and eventually burns out.

You are welcome.

And thanks for getting back to us. Now when others search for a similar problem they will see what actually worked instead of just suggestions about what could be the cause.