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Im not too sure about the fan except that the relay is on the drivers side firewall beside the battery, you might want to check it out. The coolant temp sensor for the gauge is on the passengers side of the engine between the rear 2 spark plugs. I just got finished replacing mine and my coolant temp display works great now. (It always showed "LO" before). It may have a broken wire or the sensor may be bad. You can test it by disconnecting the wire (it is a single wire with a push on connector) and connect it to ground. Turn the switch on or just start the car I cant remember which and your coolant temp gauge should read 300 if not look for a broken wire. If it reads 300 F then you need a new sensor. I got mine at autozone for about $7 . :seeya
Mine also, read low until the fan kick on then displayed 300F (which was false) turned out my "new" coolant sensor on the passenger was faulty. My aux fan would kick on but it took a long time to turn the primary fan. Replace that sensor it should help.
For right now replace the sensor only this might solve you fan problem also like it did me. A good way to check the fan is to jump the ALCL A & B terminals (both are top farthest right side) and turn the ignition to on (not the motor) and the fan should come on, if not then maybe the fan motor is kaput. Don't want you to waste your money.