Could be several things.
Staying cool only while moving hints at possibly bad fans or controller, or poor water pump circulation which could be the pump, thermostat or even clogged passages inside the radiator and engine.
I am not sure if you have those weird hydraulic fans or electric ones. I think either type is computer controlled. Usually when you turn on the AC to max with recirculate, the fans will come on while at idle. Not sure on the LS since I haven't had this problem. If there are two fans, this will usually force both to come on.
So try turning AC max, recirculate and watch the fans before and after turning the AC on.
You could also remove the t'stat and inspect it. Sometimes it will be obvious that it is broken. Othertimes you could put it in a pot of water and heat it til it boils. The T'stat should open up as it gets to boiling.
YOu could also squeeze the hoses while the engine is hot and running with you or someone playing with the throttle. You should feel the water surging thru the hoses and the hoses should not collapse.
Good Luck,
Jim Henderson