Here are the parts
Replacement parts for WHIRLPOOL LEQ9508LW0 | AppliancePartsPros.com

Here is the wiring diagram
http://www.servicematters.com/docs/wiring/Wiring%20Sheet%20-%203406661.pdf

Does it do this in both auto and timed dry modes.
The timer motor gets power from different circuit paths in the modes (Timer Switch 3). Direct in timed dry (Timer Switch 3 closed). From the electronic control (moisture sensor) (Timer Switch 3 open).

If it does not advance in either mode odds are the timer motor is shot.
Check it with a meter, if it gets 120 volts in timed dry and does not advance it is toast.

If it does advance in timed dry, clean the moisture sensor (Item 30 in Section 3). Isopropyl alcohol works well but a damp cloth also often works.
Be sure to dry the sensor when clean.

The way this works is there are 2 metal strips, if there is moisture between the strips a circuit path for a small electrical current is there. The board senses this and the unit does not advance as the clothes are still wet.
When dry there is no moisture, no circuit path, the unit senses this and advances the timer.

If there is dirt between the strips, it creates a circuit path and the unit always thinks the clothes are wet and does not advance the timer.