2002 Pontiac Montana cooling fans run but never switch to high mode when engine gets hot.

SOURCE: my 1998 pontiac sunfire is getting hot but not overheating

the fan may be bad but you can find the hot side of the plug and run a hot to the battery and see if the fan comes on. if it doesent you know you have a bad fan if it does you can run a wire to the fuse panel so that it comes on when the car does and shuts off when you shut the car off

SOURCE: 1999 pontiac montana 3.4 heat ing problums

sounds like a blown head gasket to me.  Does the coolant smell like exhaust?

SOURCE: overheating, coolant not getting through engine

try to flush the system wit presure water stright and reverse but make sure before flushing u must took the thrmostate out from housing and thermostate must be in right possion .

SOURCE: 2000 pontiac montana cooling fans

YES you will need to replace the sensor.

SOURCE: 2003 Pontiac Grand Prix SE 6 Cyl - Cooling Fan Not Operating

the fan motor could just be bad, you can test this by removing the fan and connecting a wire to each terminal and to a battery. the wires hooked up one way will spin the fan and the other way will spin the fan backwards so you cant hook it up wrong. I hope that helps you out!