I reviewed notes from car talk 2 years ago that I had posted the same issue. I did nothing at that time and it has been in the same position for the whole time. (don’t ask, us poor people lawyers have had a bad 2 years in Kansas)
what damage have I done to my car by not doing anything?
Is there something I can do myself to decide what to do next?
A gauge stuck in the cold position usually means an open in the circuit between the gauge and the temperature sender. Bad sender, connector dislodged, or corroded connector.
The easy way to check this is to run a small jumper wire from the sender connector to metal anywhere on the engine or chassis and turn the key to the RUN position. The gauge needle should go all the way to hot in a few seconds. If it does the sender is faulty.
The sender is also an easy, cheap repair. There’s always the possibility of a faulty dashboard gauge the the odds of that are very slim so I would shuffle that scenario to the back burner for now.
Do not confuse the gauge temperature sender with the engine coolant temp sensor (part of the engine management) or the temperature sensor which activates the radiator fan.
As to damage, this won’t hurt anything unless the car has overheated and it was not caught for whatever reason.
You don’t need to buy anything. A jumper is just a short piece of scrap wire used as a means to an end. You could even use an old paper clip. Just stick the end of a bared wire or paper clip into the connector from the temp sender and touch the other end firmly to metal on the car.
Might be a bit easier if you have someone operate the key (don’t start the engine) and watch the gauge for you.
Regarding the senders/sensors, the gauge sender should be a single blade connector; meaning it only has one wire.