I have a Moentrol single lever shower valve and only hot water is working. No cold water. The valve is 19 years old. What can be done short of replacing the valve?

Replace the cartridge. Once that's been replaced you have rebuilt the valve.
I have put up a exploded view of a Moen shower valve and here are the instructions on how to change the cartridge.
Turn off the shut off valves to the shower valve and remove the handle
Remove retainer clip (copper horseshoe shaped clip) and pull out cartridge with channel locks, pliers, or cartridge puller. Lube new cartridge with silicone jel and
Insert the cartridge with hole on the stem facing the 6 o'clock position, or down in a tub/shower.
Reinsert retainer clip and screw in handle assembly, followed by the cap or insert button. Turn water back on . My company uses Moen valves on all their new construction. Moen makes a "cartridge puller" That will get those stubborn cartridges out of the valve body if you have a problem. Good luck, Tom