Why is my 2001 Buick's temp gauge increase drastically while idling?

From the description of the coolant, it sounds like the cooling system needs service. But that may not be what's causing the overheating when stopped. The problem might with the radiator cooling fan failing to come on. Start the engine and let idle while at the same time watching the temp gauge. As the temp gauge rises, check to see if the radiator fan engages. If it doesn't, there'a a problem in the fan circuit. This could be a bad fan motor, bad fan relay, bad coolant temp sensor for the fan, or a blown fuse for the fan circuit.

Rafiator cooling fans working properly? .. While vehicle is parked/stopped with engine on, that's the only way it has to keep the engine cool .. At highway speeds, don't need a fan .. Ram air does the job!