I think you are on the right track. It does sound from those symptoms that there is air in the system. I was going to suggest that you flush the core in both directions, but I see you have already done that. Hopefully it is not something else that is getting air into the system, like a leaking head gasket or cracked head.
For now, I think you should try to go through the drain & fill procedure (fill really), and try to get as much air out of the system as possible. That is where those bleed screws come in. When the engine is cool, fill the system at the radiator, with the screws open, until coolant comes out of those screws. Then - while coolant is coming out, close the screws quickly. If coolant comes out of one first, close that one, then do the other one.
Also, make sure that the overflow bottle is at least half full all the time. Then, over a couple of heating and cooling cycles, coolant should be pushed into the overflow bottle, along with any air trapped in the system, and only coolant is pulled back in. This is kind of a 'progressive bleed', as I call it. It may take a few cycles of the engine heating & cooling off to get this done, and the overflow bottle has to have coolant in it at ALL times.
You should also check to make sure the area of the radiator cap is clean, and if the cap has not been replace in the last couple of years, that should probably be done too. The objective here is to get the system FULL with no air, and keep it that way.
I will also check my system for any other notices that might involve this problem and get back to you if I find anything.
Please let me know how it works out, or if you have more questions.
Take Care, Greg A.