I have a Kenmore Elite Total Care dryer that has worked fine until now. We recently replaced its companion washing machine, so I am wondering if the dryer is at end of life as well.

The dryer will run, but after a minute or so of running, it switches to the lowest heat setting and sets itself to 19 minutes. If I press the Stop button, it stops for a few minutes then the display comes on again with 19 minutes and the lowest heat setting selected.

It's been randomly turning on with the setting at 19 minutes for a few months, but at first I figured I was forgetting to shut it off. Now I see that's probably part of a larger problem.

We had the vent lines checked for excess lint last fall and were told there was nothing there, so I think that's okay.

Is it at end-of-life? That wouldn't surprise me if it is. The house is 28 years old and we bought it 3 years ago; appliances came with the house but no manuals were left, so we don't know how old they are.

All driers will do that when the lint builds up like concrete on the inside of the air ducts and the grease gets dried out in the bearings. You need to totally tear it down and scrape out and wash all of the parts except the motor, which you vacuum out unless you want to do some really expensive repairs later. Halfway doing it will not work. This includes the exhaust vent, all the way, even the outlet on the side of the house, if you have one. You can go to YouTube to see videos on how to take it apart, but do a very good job or it will fail again quickly. I would get a new belt and change it while you have it apart and get some white lithium grease and grease all bearings you find. You need to pull the support rollers apart to grease the pins with white lithium grease. You need to clean out a drier every year, at least, according to the manual. All sensors work together by comparing temperature, so all need to be cleaned. You just unplug the wires from the sensors and remove ductwork that they are attached on (Label wires!) and clean the entire thing with soap and water. Make sure you go all through the dryer and get all parts like the blower fan and housing, vacuum out motor well and clean exhaust. Mark all wires you unplug so you know where they go. This is an easier job than you would think. You need a meter to test any fuses that may be blown. You will see them on the schematic diagram.
See: Dryer vent cleaning prevents fires - YouTube (http://www.youtube.com/watch?v=A6wmD502-xE) and: Cleaning out a dryer with top lint cleanout - YouTube (http://www.youtube.com/watch?v=OBNDbyRm68M&feature=fvsr)
Cleaning out a dryer: By a woman 2 of 2 - YouTube (http://www.youtube.com/watch?v=a73HTFkcB90&feature=relmfu)
Manuals: https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20Dryers.zip
https://dl.dropboxusercontent.com/u/8010163/Kenmore%20Dryer%20Manuals.zip