Your model number looks to be short 3 digits at the start, please check it.
Perhaps if the unit has a name other than just Kenmore it will help.

Dryer shuts off before cycle ends.
If you have to wait several minutes before you can restart the unit, it could be that the motor's internal thermal cut-off is tripping. Then you have to wait for the motor to cool down and the cut-off to reset before it will start.

Dryer does heat up but dial does not move either in timed or more dry/less dry areas.
This is a puzzler; if it is the motor overheating the timer should still advance in timed dry. Usually the timer gets power directly from half the line in timed dry and this is the same side that runs the main motor.

Do you know what element needs to be replaced.
Cannot say.
You may want to open up the control console and take a look.
With a little luck perhaps the wiring diagram is in there.
Also it may be glued to the back panel.