I have an older Kenmore 80 Series dryer. The drum stops turning after it runs for a while (5-10 minutes). It sounds like the motor is still trying to run before a safety switch shuts it off. After I let it rest for a while it starts running again. The belt is still in tact but looks old but has no significant cracks. If it is the motor, how much would that cost to replace?

This does sound like a motor problem. If you can remove the front panel, it usually has two clips one on each side of the top, I do it with a putty knife. After the motor has run, unplug the washer, and feel the top of the motor for heat. It seems that it is shutting off internally because of an overload. Motors range in price, call your service department or call Kenmore, they probably have a toll free number. You will need the model number and make when calling. It may not be worth working on, with the age of your washer- they normally last 11 to 15 years depending on the usage. Good Luck!