What would cause the wipers on a 1993 Silverado to continue to run after they have been shut off and then stop in a random position?

Replace the circuit board for the wiper motor or pull the circuit board out and solder the loose connections. Our 1995 Silverado had a recall on it for this same problem, took it to the local dealer where they were happy to fix it for free as it was recalled for that. Ask at your local Chevy dealership, they might do same for you.