If your fan is working properly and make sure it is not loose and it has some tention to it when you spin it. Check that your coolant is topped off best time to do it is before you start the car in the morning when it has sat for a while. If your coolant is full then change your thermostat.
It sounds to me more like a thermostat or low coolant. Seeing as when you sit it gets hot and when you move it is normal.
