Low Coolant light keeps flashing but radiator and reservoir are full

Try a new cap if you haven't if it doesn't hold pressure it will run hot . also look for a bleeder valve near thermostat or on a pipe near valve cover and bleed all the air out of the lines . some have two valves look close .

If it is a recent model car it should be filled under vacuum otherwise air in system affects the sensor and it thinks the system is empty. Most recent GM vehicles are filled under vacuum now, I presume other makes are similar.