My cooling fan keeps on running

My cooling fan keeps on running. The heat guage is full temperature. There is nothing wrong with over heating. Battle to start. Have to unplug heat sensor to start the engin. When heat sensor is unplugged the fan still runs and heat guage on min. When heat sensor is unplugged and I push the plug back half way the fan stops. The engin won't start then. Tested the sensor when cold meausure 50 ohms. Is that correct. Is the heat sensor faulty.