Here are the parts
Replacement parts for 110.66732501 models | AppliancePartsPros.com

See the attachment for the wiring diagram.

Does the timer advance but just not shut off?

When you replaced the heater was it also not shutting off?

Did this unit used to advance/shut off or has it never done this for you?

This timer runs directly off the 240 in timed dry.
Since you have replaced timer, I would unplug the unit and check the wires to the timer motor and that timer switch 0 timer contacts are closed.

Cannot say if you can fix it yourself probably depends how motivated you are to try it.

The timer advances but does it stop advancing at the end of cycle or does it just keep advancing into a new cycle and never stops?

To check the timer switch 0 on the timer.
Unplug the unit and set the timer to just before off on the dial.
Now unplug the TM wire at the timer connections and measure across TM to OR, it should be 0 ohms.

Also could be the heater has grounded.
To check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 to 12 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.

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.
4. When you start always short the meter leads together. This will tell you that the meter is working and if there is any 0 offset.

There is a good STICKY at the start of this forum about it's use.