Engine temperature gauge show hot but vehicle is not overheating

Yes. This is what it sounds like. If the engine is not actually overheating then it sounds like just a sensor or control issue. Basically the sensor sends its signal to the PCM and then the PCM sends it to the gauge in the cluster. So this is either a sensor issue or maybe a control module issue.

The way to confirm it is to hook up a scan tool and see what the sensor is telling the PCM and if it is high then you know the gauge is accurate in what it is being told to display but the sensor is not reading the accurate temperature.

Otherwise, we will need to monitor the voltage coming out of the sensor to determine if it is accurate. I attached the replacement info and wiring diagram for your review. Let me know what questions you have. Thanks