Hi, Tref! Try this...
Do you have access to a Volt-Ohm Meter (VOM) and a basic knowledge of its use? It's not too difficult, and I'll assume you do, or have a friend who can help.
With the engine cold, and ignition switch off, find and disconnect the temperature sending unit lead. Set the VOM to Ohms and measure resistance from the connector on the sensor to ground (chassis, engine block, etc.) I don't know proper values, but something under 100 or 1,000 or so would be a good guess. More than that, like 1,000,000 or more, and the sending unit is "open" or "bad"
Let's assume you have a "good" sensor.
You can warm up the engine and take another measurement with the engine hot. The "hot" reading should be quite different than the "cold" reading.
You can also remove the sensor and take resistance measurements from the connector to the threads when it's at room temperature and after dunking it in hot water; remember it's hot when you grab it!
Sensor looks good? OK, let's try the gauge.
Set the VOM to Volts. Select at least 15 volts full scale, or the meter might go POOF!)
(Ask me how I know this sometime...
Turn the ignition on, and look for voltage from the sensor wire lead to ground. I don't know how many volts would be "good" but zero is "bad."
MOST gauges are connected on one end to battery voltage (12v, 5v, etc.) in the instrument cluster, and are "grounded" through a sensor. The sensor changes its resistance depending on what it's sensing (temperature for coolant, pressure for oil, level for fuel, etc.)
Lower resistance allows more current to flow through the gauge, while higher resistance allows less. The gauge basically indicates current, and the needle moves up or down accordingly. You guessed it! Some move up with more current, some move down.
Service manuals and electrical schematics are ESSENTIAL for troubleshooting.
Hope this works, and it's only the sensor that's bad. Good luck!
'96 E300D 60k mi (wife's daily ride)
'95 Audi 90 120k mi
'92 GMC Suburban 139k mi
'85 300SD 234k mi (my daily ride)