Why won't my AC system turn off after I flipped the breaker?
You are missing something.  Either you have not found the correct breaker yet, or the breaker is not inside the breaker panel for your house.

Walk through each breaker in the breaker planel for your house.  You could have a mislabled breaker.  (The breaker you are looking for is likely a two-pole 240V breaker, maybe 30 amps or better.)

Since this is a central cooling unit, it might be powered from its own breaker box.  Follow the electrical supply wires (cable) back from the unit.  Where does it go?  You might find another breaker box dedicated to the cooling unit alone.  There might also be a breaker in your meter box protecting the unit in addition to the main breaker for your house.