I've recently been using the C5 I inherited that hadn't been used for many years.
I'd fitted a new water pump/thermostat/hoses and Dexcool mix. I haven't anything to compare to but my car was doing exactly as you describe above (regarding it getting too hot but fans seemingly working etc). I of course wanted to try and sort things out.
I was needing a new AC condenser anyway, so got on with ordering one from Rock Auto and bought a new radiator too. My condenser and radiator were both a bit corroded due to age but I can't say either were blocked or damaged. I was actually a bit worried that my problem would persist but I'm pleased to report all well. Now cooling as expected circa 190c when moving. C5's are obviously not cars that sit steady at the temperature gauge. It seems normal that if ticking over or in traffic the temp will go up but the fans bring temp down fine if in traffic or sitting ticking over. I can see why guys buy the DeWitts double core radiators. I reckon the factory one is only just adequate particularly with the "bottom feeder" aspect.
I could soon tell my cooling was much much improved and not to be worrying about with the new radiator. I would firstly try to see if the condenser or radiator do look clogged or blocked in any way. Being that your symptoms are identical to what mine were I think you can 90% feel sure the water pump is ok (cooling down when fans come on reinforces this theory). I would then look at changing the radiator, like I did. Not in the least prohibitively expensive with many on Rock Auto around $100. I just went for aftermarket. I may have gone AC Delco but for some reason postage was a lot more (maybe much heavier packaging).
Also, I fitted the air dam back to my car. I don't think this was the only problem though (running without it) because now at tickover it's cooling much better and is also a bit slower to warm up from cold than it was. To begin with I thought it was normal that the car heated up fast. It was only when I started driving it more that I could tell cooling wasn't as efficient as it should've been.
Lastly, it does strike me as very important not to get air in the cooling system, and "burping" it. My tip whether others agree or not is to fill the system very very slowly. It was just last week I changed the radiator and I filled it slowly with a kettle, stopping and listening at the expansion tank for air. Ear to the tank between fills, air gurgling it's way out could be heard. I'm of the opinion a fast fill up of 8-10 litres would do nothing but trap piles of air.
You can undo the "steam pipe" ends (there is a metal pipe beneath the throttle body with metal unions either end going on to each cylinder head) to let air out at this high point in the cooling system but not all cars use an O ring to seal, some have a GM gasket with built in rubber washer/o ring's type thing. You'd be lucky to re use them. Dorman make replacement little gaskets, part 56390. Available from Rock Auto.
Hope you get things sorted out fine.
