My experience is that when a display unit is consistently 20 degrees off of actual temp, it's a bad unit MOST of the time. The temp sensor is a simple resistor cable that mounts next to the front license plate, and is in the windstream, and there is honestly little to go wrong with it (though they CAN go bad). As the
@gsxr said, you can diagnose both.
I would consider acquiring another display unit. Unfortunately I used to have masses of Fahrenheit units that I would get in the wrecking yards, but the supply has completely dried up over the past two years, and one hardly sees any 124s in the wrecking yards anymore.
I do have about 10 spare W126 temp display units in my parts stock, available for sale, but they are unfortunately physically larger than the 124 units, and also incompatible from an electrical connector perspective.
Also, there were two versions of the W124 units that were made - a first version for W124s from the 1986-1989 time frame, and ones from 1990 through 1995. They differ in capability as well as in the electrical connector on the rear. You want to get a spare unit from one of the 1990-onward W124 cars.