P0128 Diagnostic and Fix for 2008 Buick Lucerne - Engine Coolant Temperature Sensor Issue

I'm experiencing a P0128 fault code on my 2008 Buick Lucerne. Upon starting the engine, the dashboard displays 'Engine Hot A/C Off', which is unusual because it's currently 0 degrees outside and the vehicle has not had time to warm up. The engine temperature gauge remains at 'C' (Cold) even after allowing 20 minutes of idling or driving between 25 and 45 MPH. Normally, once warmed up, the gauge settles around halfway between Cold and Hot. However, this behavior persists despite normal operation. The cabin heater functions but heats up slowly due to external conditions—still within reasonable time. Coolant was last topped off during my recent oil change, which occurred 2,000 miles ago. I'm seeking advice on the following: - Should I pay for a $100 diagnostic test or simply have the thermostat replaced? - Is it safe to drive the vehicle daily (maximum 3 miles at speeds up to 45 mph) until I can visit a mechanic? Any insights into whether this is a sensor issue, thermostat failure, or another component would be greatly appreciated.