I do not know if I can be much help, but here goes.
I do not have a wiring diagram for this unit.

Sounds like there may be 2 problems.

[COLOR="Red"]Not Advancing[/COLOR]
[COLOR="Blue"]I understand that there is a moisture control board, but have looked at the schematics and found none for this model. Could it be the dryer cycling thermostat or the thermal cut off?[/COLOR]
I doubt it, they control heat not the motor. There is a thermal fuse in the motor circuit but if blown the unit will not run.

There probably is not a moisture board.
The way it works is that in timed dry the timer motor is connected across one side of the line (120 volts).
In auto modes it gets it's power through the heating coil when the coil is turned off by the thermostat. There is then a resistor which drops the heater's 240 volts down to 120 volts for the timer motor.
Often no timer advance is a blown resistor or a grounded element.
Resistors are usually around 5,000 ohms.
For 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 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 so it never shuts off and you never get a timer advance.

[COLOR="Red"]Will Not Shut Off in either mode.[/COLOR]
This has me sort of stumped I would have bet that replacing the timer would have repaired this as there is a set on contacts in the timer that control power to the motor.
Did this unit ever run correctly or is it new to you, just wondering if the wiring is messed up.
Could be a defective replacement but it would be very strange to get a timer that has the exact same problem as the original one.
I am assuming that that opening the door stops it and the the push to start switch operates correctly.
If you manually move the timer to off, does it stop?

I checked the coil. It's fine also. I just can't figure it out.

Any suggestions?

I have attached the wiring schematics. I think you need to click on the attachment (paper clip) link at the top.
I will say that I have owned this dryer since new and have not ever had any problems with it. In the timed dry mode, it will advance right to where it should shut off and continue to run at that point. At that point it will be in the cool down mode, the last ten minutes of the timed dryer mode the heater turns off, like it should..

Hope this helps.