dryer runs fine only if you hold the start knob
Here are your parts
http://www.appliancepartspros.com/partsearch/modelsearch.aspx?model=gtdp300em1wws
Sounds like a bad motor.
The motor has two centrifugal switches which activate when the it gets close to operating speed.
One of then closes and connects in the heating coil. This ensures that the heater does not come on before there is air flow.
The other one has two functions.
It open a set on contacts and disconnects the motor’s start winding as leaving this on all the time would overheat the motor.
It also closes a set of contacts and provides a circuit path which is in parallel with the start switch. This lets you let go of the start switch and the motor continues to run. It sound like this is not working.