2005 not running properly

I have a 2005 Aveo that is not running properly. A few years back I had to change the thermostat housing as to the old plastic one breaking, I was fortunate as to the motor not blowing as I had seen with others. The car was overheating and losing antifreeze and I finally investigated and found the problem and fixed it. Since I have had coolant problems, the car is still losing antifreeze/water as was before but I do not see any leaks. I notice when the coolant gets low it starts to overheat. I am constantly adding antifreeze/water. I took off the fill bottle cap and steams comes out which tells me it is still under pressure. I have checked the oil to see if any water appears in it and it seems to be of a regular color or make up. The odd thing now is the car is starting hard and running funny, not sputter/sputtering but sputtering some. I am concerned it could be the head gasket but the oil does not indicate any liquid going there. I hate to be long in my writing but trying to give the best feed I can to get the problem analyzed. I would appreciate any help, thanks and God bless.

Hello!

My first thought is the throttle body. Antifreeze goes through it to prevent it from freezing, and so it may be possible that there's a leak in there.

You mention you changed the thermostat+housing, have you done the rest of your regular maintenance? If you look in my garage in my profile, you'll see the bare minimum you should have done for your car's age (I'd do more with mine, but working minimum wage only goes so far at a time!)

From what you wrote, it sounds like a blown head gasket, and a block combustion gas test (or leak down) should be able to confirm that for certain. Can you clarify the timeframe on this coolant loss - it sounds like you're saying it's been losing coolant ever since the T-stat replacment, but maybe I'm just not understanding that correctly.
If it's been losing coolant for a long period of time, that could be worse than the HG problem which it likely has. The reason is that coolant would have been washing down one or more cylinders and diluting the oil, which is a really bad thing. If the bad HG is confirmed, and this has been going on for a while, you need to consider your strategy for this vehicle. If you want to keep it for an extended amount of time, I'd urge you to consider a rebuilt engine (you can buy them with 1 year warranties). The HG repair alone will be 1,000 or more, and will do nothing about the coolant wash issue that's been going on. The other possibility would be to get an engine from a boneyard, but that's a real crap shoot, given the timing belt failure rate on these engines.