Here are your parts
Replacement parts for MAYTAG MDE9206AYW Dryer - Ele | AppliancePartsPros.com

Here is a manual
http://www.servicematters.com/maytag_library/docs/16023086.pdf

It could be a number of things.

That it will not do a cool down at the end of a cycle points to a grounded element as this function is controlled by the timer and not a thermostat.

Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 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. This can cause it to run on high and the thermostats cannot regulate it.

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.

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