why is my 2000 chevy malibu overheating?

[heythereamberr]

Last winter it was overheating and the heat didn't work and I got a new engine. Problem solved. In the summer I got my a/c fixed and it began running hot again. We took the thermostat out and it ran fine with no troubles for a couple months as long as it wasn't driven or about twenty miles or so at a time and the a/c is not on. We just put a new thermostat in and it is running hot again and the heat doesn't work. It starts to cool down some as long as there is no kind of air running. What could be the problem? Maybe a heater core? Also when the radiator is flushed gritty black stuff comes out like maybe stop leak a previous owner had used. Also there is nothing leaking out of the car.
Yes we never used it. But it comes out everytime the radiator is flushed.