Dryer runs on heat settings for awhile and then shuts down. It will run for full cycle when heat setting not selected?

it is most likely the motor overheating. it is a very common motor, and reasonably inexpensive. the part # is 279827 tom

Is probably some heat limit switch?
It is a possibility,
There are only 2 heat sensing devices in the motor's circuit.
First the THERMAL FUSE but if it blows it does not reset.
Second is a thermal protect in the motor itself, it will open and reset when the motor cools down.
The other thermostats for the heater should have no effect on the motor so you would loose heat but the unit would not shut off.

What is strange is that the temperature setting has an effect on the unit shutting off. It should not have any effect on the motor.

Dryer will not restart immediately unless a no heat setting is selected and then it runs fine and for the whole cycle.
So if it has stopped and you put the unit to no heat the motor starts right away. But if you leave it on a heat setting it will not start right away.
Is this correct or have I misunderstood the above?

If the above is correct then I will have to sit down with the wiring diagram and try to figure out a very weird problem.

I think I would check the motor first just to make sure it is not clogged up with lint (a long shot)

My thinking right now is that, it is a house wiring problem.
This is only applicable if it is the motor overheating.
It is a possibility that the line is not providing enough power, so it runs OK when just the motor is running, once the heaters start, the voltage on the line drops due to current draw.
The low voltage causes the motor to overheat, tripping it's thermal cutoff.
Only way to test this theory is with a meter.
Check the power L1 to L2 should be 240 volts.
Then check it while running with the heat on, should stay close to 240 volts.

Do not know your background so thought is should add
Be careful 240 volts is lethal!!!