I believe the temp sensor is a three-wire type.
One terminal for the gauge, and the other pair for the PCM.
You could measure the resistance across the two PCM terminals and hopefully read a high resistance when cold and lower resistance as the engine gets warmer.
The remaining terminal is for the gauge and resistance would be across the terminal and sensor body/engine block/ground. Same action here, cold, high resistance, warmer temperature, less resistance. No action happening with either circuit, new sensor needed.
You could unplug the connector and jumper in a 0-1K potentimeter/variable resistor to the gauge lead and ground, ign on, and vary the resistance and view the gauge for sweeping action. if gauge works, new sensor needed. If gauge does not work, bad gauge or connection from the sensor to the gauge or loss of power to the gauge.
However, if the engine fast idles when cold, then slows down after warm-up, it is a good bet the fans would come on as the PCM still works.
Have a new sensor in hand. Sensor should/might have sealer on the threads.
Optional to apply one lap of Teflon tape to threads.
Engine cold, remove the sensor in one hand, and quickly replace with the other hand.
The faster, the less coolant lost.
The reason for the optional thread tape is the sensor body should have a good contact with engine head/ground to have a complete circuit for the gauge.
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