If the things related to flushing the radiator (that Schmoe mentioned) are not working....
You might suspect the head-gasket. Classic headgasket symptoms are overheating with loss of coolant, or overflowing the coolant overflow bottle without an overheat condition. A bad head-gasket can be verified at a decent radiator shop. They have a little device that they hold over your hot open radiator and it "sniffs" your coolant. If the chemical in the sniffer turns blue, then you have hyrdocarbons in your coolant, which indicates a bad head gasket. This is how I diagnosed a head gasket problem on my '69 Bronco project.
Next time you dump the oil, see if it's milky looking. Some GM's suffered from intake manifold leaks. (I had this problem on my '98 Chev K1500.)
When I have a problem with one of my vehicles, I usually do an internet search and find out what the common problems are for that car.
As my dad used to say..."common things are common". There's some fancy philosophy term for this expression and it sounds more egg-head (I think it's called Ockham's Razor..I digress). Anyhow, if you can find out the problems people are usually having with this car with the same symptoms, then in my experience you've got a pretty good chance of having the same issue.
Good luck with this problem and let us know what the problem turns out to be!