I'm guessing you have a thermostat problem, and it sounds like it's not opening properly or as much as it should. It doesn't matter when the fans turn on if there isn't enough coolant flow through the thermostat to remove the heat with the radiator. The fact that the guage reads correctly when the car is moving tells me that the gauge (and coolant temp sender) are working properly, and you've also replaced the fan switch. So it sounds like the fans are probably working properly but there isn't enough coolant flow to the radiator to remove enough heat when the car is not moving.

Thermostats are cheap and easy, and it isn't a bad thing to replace simply for peace of mind.