Engine Cooling Fan fault, 2000 Impala LS

My wife owns a 2000 Impala LS 3.8L at 141,000 miles for starters. Now, we have a problem with overheating,
which was recently solved with the replacement of a thermostat, but G-damnit the cooling system had to
fight us yet again.

The engine cooling fan is now the problem. the passenger cooling fan used to make scritchy-squeaky abnoxious
noises during the summer, but when it shut up finally in the fall, we thought that was just a passing annoyance.
Now the motor won't work at all, and the driver's side motor won't start automatically when the temp needle gets
above half.

-I then checked all the fuses in association with engine fans and the like.
-I replaced the master relay for the two fan relays
-I replaced the passenger-side fan relay.
-I noticed how the A/C is the only thing that can turn on the driver's side fan, under any condition, so
I tested the possibly broken passenger side fan by switching the two fan's power supply cables.
-The result is neither engine fans would come on because of high engine temp (expectable)
-The driver's side fan came on with the A/C switch.
-The passenger side fan did not come on with the A/C switch.

My determination is the passenger side fan motor is bad. Unfortunately I can't switch it out, because fan motors
come in sets of two in a single housing unit that comes for $140 minimum. I figured that I would rather use-up the
remaining good engine fan for both A/C and cooling purposes, so that leaves me with one goal:

I want to find what controls the auto-on switch for the engine fans when the temp gauge gets too high for
normal operation. The thermometer works just fine, else I wouldn't know whether the engine were too hot or not.
Where oh where. Does anyone here know what I should replace/test next?