To elaborate. As mentioned, the fan "relay" is not the mechanical relay of earlier vans (80-90's). It is a solid-state device, using transistors, very similar to the cabin blower "resistor" (also transistors) used in vans with the Automatic Temperature Control (digital display). When solid-state devices fail, they usually fail shorted, so not surprising your bad one was running with the engine off, since it always has 12 V power.
Many here have experienced "blower fan always on high" with ATC, as I did. Since that "resistor module" costs $60 I tried fixing mine by replacing the drive transistor (MOSFET I recall). I even put in a slightly better one (lower on resistance) and cut slots in the cooling fins. It worked for ~1 yr then "blower always high" again. I finally bought a new module with the round pins (new design).
I haven't fooled with the radiator fan controller and it looks like it is epoxy-potted (hard to repair). But, just like the blower controller, it needs a good heat sink, as was mentioned here, and it doesn't have fins like the blower controller, but relies on heat conduction to the bumper. Always install tightly with thermal grease, and sand away any rust. My guess is they use "pulse width modulation" to control the motor speed. That switches the current on & off fast (~30 kHz) and the motor averages that. It is the simplest way today to realize variable speed in a DC motor.