Here are your parts
Parts for Amana AE4213 Dryer - AppliancePartsPros.comI cannot find a wiring diagram for this unit so can only give you general info.
The wiring diagram may be stored in the control console of the unit or glued to the back panel.
On most units the motor has two centrifugal switches.
These switches close when the motor gets close to operating speed.
One switch connects in the heating element to ensure that the heater does not come on till there is air flow.
The other switch has 2 purposes.
It provides a parallel circuit for the start switch so that you can release the switch and the unit will keep running.
It also disconnects the motor's start winding as leaving this on will cause the motor to overheat. Note that the motor has an internal thermal protect which opens if the motor over heats. You then have to wait for the motor to cool down and the protect to reset before it will start again.
It sounds like the switches are not being activated.
Could be that the motor is clogged solid with lint etc. and is preventing the switches from activating. May just need a cleaning.
Could be that the centrifugal mechanism in the motor is broken. You will have to replace the motor.
Could be that the switch is bad. You will have to replace the motor.
Since you have to open the unit up you may as well give it an overhaul.
Check the belt, rear drum support rollers, front drum support glides, idler wheel, all internal seals etc.
Give the interior a good cleaning including the blower wheel.
Also this would be a good time to check/clean your vent system.
If you have to replace the motor I would not order any parts till you get the blower wheel off. It is usually screwed onto the motor shaft (left hand thread) and often has seized onto the shaft so must be destroyed to get the motor out.
Be sure to leave all wires connected or make yourself a diagram as to where they go. Also of couars unplug the unit when working on it.