I'd change the switch, 10 or 15 bucks is cheap enough. If you're still having an issue start checking the rest of the cooling system, radiator(how old), waterpump clearance from the divider plates to the impeller...etc. If you have access to or want to buy one, an infrared thermometer gun is good to have to diagnose these issues. Check the stat housing when you start the car and you know it will trigger the light, what's the temp? Check the radiator where the top hose goes in, should be within 10-15 degrees of stat temp rating, then check the rad at the outlet at the bottom, a good drop of about 40 degrees or so should be observed. Let us know what happens.