Good response to flush the system. Dex-Cool (Dex-Mud) doesn't mix with other types of antifreeze. Most people elect to get that stuff out and put in regular antifreeze. GM advertised it as "lifetime" coolant to sell more cars but right on the overflow was a sticker that said to replace it every three years. Even the antifreeze manufacturer doesn't say it's lifetime antifreeze. Because of that advertising a lot of people never changed their coolant which lead to that Dex-Mud nickname, acid buildup in the system, and more than normal corroded head gaskets.
A lot of manufacturers were supplied with radiators that have thin plastic that cracks when the petcock is screwed back in. That causes a slow leak. Rather than worry whether that applies to a specific car, I prefer to never open that radiator drain. Pull the lower rubber hose off instead. I also remove one heater hose from the engine. Most GM cars use plastic tubes on the heater cores that are easily cracked and broken when trying to pull the hoses off. That's why it's safer to remove a hose at the engine. Run water from a garden hose into the hose going to the heater core and into the fitting or metal tube the hose came off of. Fill the overflow reservoir too and let it drain into the radiator or engine. If there's a bleeder screw on the engine, typically near where the upper radiator hose attaches, open that too. That will help the old coolant run out from above the thermostat.
Once the water is running clearly, allow as much as possible to drain out, then reconnect the hoses. There is always going to be some water trapped in the passages of the engine block so it isn't a good idea to mix the new antifreeze and water before pouring it in. It's going to be diluted by that water still in the engine. Pour straight antifreeze in first until there's enough to mix when you run the engine. Use an antifreeze hydrometer to check the freeze temperature to determine if you need to add more water or more antifreeze. A lot of GM cars have the "radiator" cap on the reservoir which makes it necessary to add to the reservoir instead of in the radiator. That makes fine tuning the mixture very difficult. Those reservoirs hold over a half gallon so if you just fill the system enough to allow driving without overheating, the water and antifreeze will mix over a period of a few warmup - cool down cycles. At that point you can accurately measure the freeze point and add to the reservoir.
Remember to watch the dash gauge or light for signs of overheating. That "new" problem will only occur if there's an air pocket by the thermostat. Thermostats don't open in response to hot air, only hot liquid. There will usually be a bleeder screw on or near the thermostat housing or there will be a sensor or threaded plug to unscrew to burp the air.