Overheating issues

Alright, here it is. I recently bought a 99 Lumina 3.1L and drove it about 30 miles before it overheated. First thought was fans weren't kicking on so used a jumper and ran them direct and then tried again. Overheated again, so decided to try the thermostat and radiator cap. Still overheating, so I towed the vehicle home 190 miles(spur of the moment purchase, won't happen again that far from home).
So I decided to replace the water pump and back flush the radiator(flowed good), still overheating. My last thoughts are a cracked or warped head, by the looks of it somebody had recently put a new head gasket on and I thought they might have just replaced the head gasket without actually fixing the problem.
Car runs excellent, no misses and plenty of power. Not getting any mixing of fluids and no noticeable steam or water out the exhaust. This is what has me confused. A certified mechanic claims the catalytic converter could be burning the water/coolant off not giving any signs of coming out the exhaust... using plain water till problem is found or fixed.
It is overheating and pushing the water back into the overflow container then out on the ground. Of course I am not 100% sure I am bleeding the cooling system properly. I fill the radiator up & the overflow container, then run engine till it gets warm. Shut down and let it cool completely. I then top off the radiator and add more water to the overflow container if needed and then run engine again till warm again and shut off till cool once more. I do this a few times to make sure the system is full. It will blow good heat from the heater as well.... Till it overheats then I get no heat from the ducts.
Cracking open the bleeder valves only produces steam when the engine is overheated. Guess I could pull spark plugs to see if any evidence of water on the plugs?? Also the fans do kick on by themselves but only after the engine is already overheated, my thought on that is the engine is already steaming inside and the sensor(s) is not getting the proper temperature reading. On certain vehicles some sensors will only read the physical coolant temperature, not steam so I thought that could be a reason the fans kick on to late.

Some thoughts and ideas would be nice and if I am doing something wrong when bleeding the system let me know. I have different heads to put on and a gasket kit already but if it is not cracked or warped heads I would like to know what to do to fix the problem(s).

I think you nailed it. It sounds properly bled. Assuming that, the push-back of coolant to the res is almost certainly compression in the cooling jacket. That's also forming the air pocket that renders the t-stat and fan sensors useless...

Usually when that's the case, if you run the radiator cap cap loose and remove the t-stat, they'll run a decent temp but continue to push water out until about half full.....usually a combustion chamber hairline crack in the head is the cause. I think you're right, the P.O. missed the point and only replaced the head gaskets...

As I thought. A few really good choice words to myself for buying another POS that I have to work on. Couple people I spoke with in person said a cracked block but I have my doubts about that.
I do have one other thing I failed to mention that is also happening. On the exit side of engine, the radiator hose will sometimes collapse. I plan on replacing that but thought I would mention it.

A lot of if not all Cadillac Northstar engines overheat like this. I had a 97 deville did same thing. Threads in the block were bad so i junked the car. made no sense to spend a lot of money on a 10-12 year old car.

Got plans to work on it Thursday and/or Friday( I am a slow worker). Weather supposed to be decent then plus I have the car at a friends house about 40 miles away. Where I live I don't have the room to work on the car without neighbor griping and complaining.