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See the attachment for the wiring diagram.

The cycling thermostat should not cause this problem.

A common cause of this is 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.

You may have 2 problems.
One being a grounded element but at the off timer position the power should be cut to the heating element.
At off the L1 side of the line should be disconnected by the Timer Switch 2 (open at off)
Also the L2 side of the line should be disconnected by the centrifugal switch on the motor. This closes when the motor gets close to operating speed and ensures that the heater does not come on without air flow. When the motor is off it should be open.
Since timer switch 2 is the easiest to check I would start there.
Unplug the unit and the wires to timer switch 2, then measure across the contacts they should be open (infinite ohms) at the off position.

A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.
4. When you start always short the meter leads together. This will tell you that the meter is working and if there is any 0 offset.

There is a good STICKY at the start of this forum about it's use.