A Model Number might help so others can try and find a wiring diagram for this unit

Here is your parts breakdown
Model LGR4634JQ2

Here is the wiring diagram
https://www.servicematters.com/d...0Sheet%20-%208528189.pdf

Only thing I can see that may cause this problem is it the Motors centrifugal switch has a problem.

On the wiring diagram it is shown in the motor off position.
So the current path when you push start is through the DOOR switch , START SWitch , centrifugal switch 5M to the motor's MAIN and START windings.
One the motor starts the centrifugal switch, switches out the START winding so the unit continues to run on just the MAIN winding.

Now if the switch is stuck in the run position or it is mis-wired the current path becomes DOOR switch , centrifugal switch 6M to the MAIN winding only. The motor then will just hum because it needs the extra power from the start winding to start rotating.

Usually thermostatically controlled just means that there is a thermal cutoff in the motor so that if it gets too hot the cutoff will open to prevent damaging the motor.

With the start button disconected I have power to the motor when I turn the timer on. How do I test the timer? can I jump so wires?

is the centrifugal switch on the motor?
Yes

Is there a way to test it?
You could check it with an ohm meter. Remove the wires from the motor. Then check from 4M (should have had a blue wire going to it) to 6M (should have had a black wire going to it). It should be an open.

With the start button disconected I have power to the motor when I turn the timer on. How do I test the timer? can I jump so wires?
Not sure what you are trying to do?

1. Trying to check timer to see if it has a (voltage drop) bad contact.
I think that if you had a bad contact it would remove power to the motor.

2. what would cause power to motor with start button disconected?
I gave one possibility in my earlier post though that seems unlikely since both the original and replacement motors have the same symptoms.

Do you have a meter so you can measure the voltage to the motor.
It could be that one of the wires going to 5M on the centrifugal switch is shorting to ground/frame. You could check at the plug between Neutral and ground it should be a fairly high resistance.

Just thought of something. Disconnect the ground from the line cord where it comes into the machine. Should be a green wire that is connected to the frame, make sure it cannot touch the frame. Then plug the machine in if now the motor does not have power to it then a short to the frame is a high probability.
Do not touch the machine when plugged in, if for any reason the frame is live you could get a shock .