Here's how my '97 fans work.
One of the two fans will come on with the A/C no matter what the coolent temp. (I think it's the right side, passenger. EZ to check)
The other one comes on when the coolent gets hot enough. So to get both fans on takes hot coolent and the A/C. (this is in my owners manual too I think)
So it sounds like the problem could stem from a non-functional thermostat or relay. (not the thermostat that opens to let coolent into the radiator, the one that closes an electrical switch to enable a relay to run the other fan)
Heck if your lucky it could be a loose wire.
Maybe the '01 is different, but don't blow $200 + until you consider this information. :-\
Good luck,
Mark