[Lawrence]

Dryer heats, tumbles, runs, but timer won't advance. Dryer won't stop automatically.

[denman]

Here is a link to your unit
Whirlpool EED4300TQ0

Looks like all you have is the timer and the push to start switch. Does not have a separate temperature switch.
Is this correct?

Have you tried different cycle types usually timed dry uses a different set of timer contacts than permanent press etc.

Open up the control console. You may find the wiring schematic here.
Open it carefully because you do not want to pull wires off the timer, also shut the power off when you do this
Can you hear/see the timer motor running
It you can then it is definitely the timer, your internal gears are stripped.
If not running, look for loose or overheated wires

If you have a meter and the schematic you could check the contacts

Whenever I work on my dryer I clean and check my vent system. It is just good practice as venting causes many, many, many, many problems.

[kayakcrzy]

Lawrence, just replace the Timer, and you are done. If you need me, you can reach me at my web site at ApplianceEducator.com Tom