Pegged over 260, coolant temperature sensor not working properly

What does the gauge do when you just turn the key on? It should cycle across the gauges and then stop. If the gauge isn't moving it isn't the sender or a bad ground, it would be a fault in the gauge or it's drive motor. If it does cycle but then returns to that position it could still be the stepper motor. A way to reset them involves gunning the gauge test function on a dealer level scan tool. However that just does the same thing as turning the key on. Please try that and report back and we have a starting point.

It sounds like it could be an open connection causing the issue. There are 3 wires for that sensor. One sends the signal to the gauge, another is the signal for the PCM and the third is the ground. The likely suspect would be a failed ground as it would cause a problem with both while a failed wire to each of the others would cause problems with just that item.
The ground signal is shared with the MAP sensor. To test it you can use a multi-meter set to ohms. Then look at the pin with the orange wire with black stripe. This is the shared sensor ground. Go to the MAP sensor and unplug it and test for continuity between the two orange w/black wires. Then check between them and a good ground. If you find no connection then it is likely that the connector or wire has failed. If you find a ground at the MAP but not the ECT then it is in the wire that goes between them. A quick test there would be to pierce the wire right at the back of the ECT connector and test for ground, then run a jumper from that to a good ground. If the gauge corrects itself with a cold engine and key on you need to repair the ground wire.

https://www.2carpros.com/articles/how-to-check-wiring