My dryer keeps shutting off mid cycle and the timer continues to run. It seems to be some sort of short somewhere, because it starts fine as long as I'm holding the timer knob, but when I let go it stops, after a few attempts I can get it to stay on while I let go and complete the dry cycle.

If you have an outside vent, check to see if it is clogged. If clogged, this is the reason it is shutting off mid cycle.

It’s dirty and lint deposits have built up, as it always does, in all the air ducts which cause the hi-limit to trip. A little vacuuming out does nothing. You will have to take it apart and clean it really well or it is all for nothing. Every air duct will have to be removed and scraped and washed out to remove baked on lint deposits which is like a concrete lining. This will burn so the manufacturer has installed the safeties to keep from having a house fire. The safeties are oval-shaped with a round center part containing the safety with two plug-on terminals for the wires and when they burn open or trip, from the dryer being dirty, nothing will work. Yours is going off on hi-limit which cuts it off at 250 degrees. Using an ohm meter you put a lead on each terminal of the safety, with one wire unplugged, and if it's good you will have a reading of about zero; if it's bad it will read infinity. Driers were the number one cause of fires at one time, hence now we have safety fuses. Apparently yours is resettable. Look at: How to clean your clothes dryer. How to make it hot again. - YouTube to see how. Also: Mr. Appliance Tips: How to Clean Your Dryer - YouTube ; Cleaning a Dryer Vent and Lint Trap - YouTube
Electric Clothes Dryers
Whirlpool Dryer Model LE5800XSW3 Wiring Diagram
When you finish cleaning the drier it is best to also replace the belt, too, because you do not want to have to disassemble it all over again and if the belt is over two years old you should replace it.

Note that on most of the videos it does not show them washing out the parts, but this is important because it's the only way to really get the parts clean, just don't get motor wet or the capacitor if it has one. Vacuum out the motor to remove all dust. I would move the drier out to your garage to have plenty of space to work. You also want to check your vent and clean it, also. The vent should have a short run and be straight as possible. The job is really easier than it sounds. There is no shortcut that will work; it has to be cleaned well. Before you re-assemble it make sure to use white lithium grease (high-temp grease) and grease all bearings. DrTom4444