2003 Jeep Cherokee cooling fan will run until the battery is dead?

Inspect and test the hydraulic cooling module at right front of engine compartment and also the radiator fan relay. see below

Your cooling fan relay is sticking on which is right below the right headlight, here is the location and I found the cooling fan wiring diagrams so you can see how the system works with the relay location in the images below. It is not a typical relay, but more like a solid state speed control. Check out the images (below). Let us know if you need anything else.