The thermostat gauge registers way above overheating & stays there. The coolant is full & the light does not come on, the oil is fine, the fan is working & the hood is cool to touch. Could it be the gauge itself? The car has close to 140,000 miles.

First, try replacing the Engine Coolant Temperature (ECT) Gauge Sending Unit. Apply liquid gasket to the threads. If that doesn't solve the problem, the gauge is likely at fault.