I have a 2000 Dodge Neon and the temperature gauge is going up and down with a series of beeps and the check engine light remains on. The engine is fine and not overheating at all! Could it be the coolant temperature sensor causing both the gauge and check engine light issues??

What codes do the ECU show?

Remove and clean the connections going to the Temperature Sending Unit, not the ECT Sensor. Spray connection with WD-40.

Are you sure the engine is fine? It sounds like there may be an air pocket in the cooling system.