How do you fix windshield wipers on an 04 Avalanche that won't shut off?

Change the wiper motor. There is a cam on the output of the wiper gearbox ( part of the motor)

that shuts off the wipers when they reach the parked position. Yours is worn, never opens the internal switch, and the wipers just keep going.

To just make them stop, unplug the electrical connection on the wiper motor. The motor is on the firewall, above the engine, and not too far from the brake fluid reservoir. It's mostly aluminum colored, and about the size of a burrito. Probably four wires on the plug.