My temperature gauge stopped working and the cooling fan now stays on.

Cooling fan will stay on if you unplug the coolant sensor. The temperature sending unit controls the gauge.Some vehicles use the same unit for both coolant sensor and temperature sending unit. Unplugging the sending unit will make the gauge go all the way to hot.replace the sending unit and see if that will solve your problem.

if it over heats when stationary but cools down when moving , that is an indication of a cooling fan problem
If the engine has a viscous fan hub ( fan clutch) replace it as it is not spinning hard enough when stopped to draw air through the radiator
If it has electric fan/s check why it is not working at engine temperature --could be the coolant temperature sensor ( not gauge sender unit) that is faulty and not telling the ECM to turn on the relay for the fan/s or a bad fuse or relay that is not completing the power circuit

Coolant temp sensor...

If you think something wrong with relay or relay connection, check it out. Some relays in fuse box may be interchangeable, so you can swap relays to see if problem follows relay. I'd have to look at relays.

Have you checked your cooling fan to see if it is working propely? Let the car idle and get warm and listen for the fan to kick in (if they are electric) or just watch to see if the fan starts to spin faster if it is the old school type with a clutch. Have you replaced the thermostat? A bad one will let the temp rise since it opens too late and then the temp will drop off. But it won't due this only at a light and it will do it infrequently.

firstly you need to check the coolant level and make sure there are no air bubbles in the cooling circuit
secondly the thermostat could be damaged and stays in the closed position
thirdly the cooling radiator could be clogged and it must be flushed or replaced
you have to take into accont that it could be a faulty message sent by faulty temperature sensor or bad connector of the senser
so you need to open the hood and decide, is the engine very hot or is it normal but still the fan is on

This is typical of imports to cool the motor down when the car is not moving,as no cool air is entering the engine bay

Check your cooling fans to insure they are working to cool the radiator...

I’m happy to assist further over the phone at https://www.6ya.com/expert/joe_a3369546f8067589

cooling fan has a relay. check it first. Then check the ECT, Engine Coolant Temperature Sensor. It is on the engine and sends a signal to the PCM, Engine Computer.
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Section 03-03: Engine Cooling 1996 Aspire Workshop Manual
DESCRIPTION AND OPERATION Cooling Fan, Electric WARNING: WHEN WORKING IN THE ENGINE COMPARTMENT, STAY CLEAR OF THE FAN BLADE. BECAUSE IT IS ELECTRIC AND TEMPERATURE SENSITIVE, IT CAN START AT ANY TIME IF THE IGNITION IS ON.
The electric cooling fan is an electro-drive type. Its operation depends upon engine temperature and ignition switch (11572) position. If engine coolant temperature reaches 97°C (207° F) and the ignition switch is in the ON position, the engine coolant temperature sensor (ECT sensor) (12A648) sends a signal to the powertrain control module (PCM) (12A650) which turns the engine cooling fan motor (8K621) and fan blade assembly on. Because it cannot operate unless the ignition switch is ON, the engine cooling fan motor and fan blade assembly does not operate after engine shutdown. The fan control relay (FC relay) is located in the LH front corner of the engine compartment, between the battery (10655) and the headlamp assembly. Circuit protection is provided by a 30A fuse labeled COOLING FAN in the main fuse junction panel to the left of the battery. If the vehicle is equipped with air conditioning, an additional relay is installed in the circuit. The A/C clutch control relay (19D572) bypasses the engine temperature portion of the circuit. The bypass circuit allows the engine cooling fan motor to operate whenever the A/C switch is engaged.
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Fuel pump/sender access is through cover under rear seat.


Fuel Gauge

1. Remove the instrument cluster. Refer to the procedure in this section.
   

1. Using a jumper, connect the battery positive terminal to the fuel gauge "+" terminal and the battery negative terminal to the fuel gauge "-" terminal.
   

1. Connect one lead of a Rotunda Instrument Gauge System Tester 014-R1063 or equivalent to the "F-U" terminal of the fuel gauge (9280), and connect the other lead to the "-" terminal of the fuel gauge.
   

1. Adjust the instrument gauge system tester to the resistances shown in the illustration.
   
   
   
   
   

1. If the fuel gauge does not operate as specified, replace the fuel gauge. If the fuel gauge is OK, return to the Pinpoint Test.
   

Fuel Level Sensor

1. Remove the fuel level sensor (9275). Refer to Section 10-01 for the removal procedure.
   
2. Measure the resistance between terminal A and terminal B of the fuel level sensor.
   
   Connector Shown From Component Side

1. Adjust the fuel tank float to the following positions:

1. If the resistances are not as specified, replace the fuel level sensor. If the fuel level sensor is OK, return to the Pinpoint Test.
   

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Cooling Fan Malfunction. Your radiator fans are most likely not working. That's why the temperature is OK when you're driving full speed and the wind is cooling your radiator. You need to check and/or replace your cooling switch module and radiator fans (most likely it's the cooling module). If you were sitting idle or in stop-n-go traffic for an extended period of time, your car WOULD overheat because the fans aren't kicking in cooling your radiator - then you would see all of the cooling system water spill out of the coolant reservoir, before this would happen your temperature gauge would go to the red and the digital dislpay would flash a thermometer (indicating you're about to overheat in 2 mintues). I hope this helps someone out there - Yes, it happened to me in the middle of nowhere.