Engine light comes and radiator fan runs full speed from start of engine. Water is full. Dash say turned off air conditioner because of engine over heat. Temperature gauge reads cold.
Chevy used to have two temp sensors on many of their engines. One is sender for gauge, second is switch to signal overtemp to ECM. I’d look to see if that way on yours and check switch op with a meter to verify.
Gauge can read cold on overheat if coolant low in engine. Air pocket by sensor doesn’t work right. You say water full but are you just checking expansion tank level or also radiator fill level?
It appears to have a pressurized reservoir:
Are you actually using “straight” water in the cooling system, instead of the necessary coolant/water mix?