The overall movement of an object or gas is not the same as the ‘wiggle’ of atoms within that object or gas.
Cooling in an AC means reducing the wiggle. Often this wiggle-reduction is not induced by any sort of slowing or even cooling in the typical (heat exchange) sense, but by compression.
You can look up explanations of how fridges work if you’re curious about that part.
So yes, the atoms are slowed, but no, the air as a whole is not slowed.
When people say that higher temps mean more movement they don’t mean movement in a traditional sense. I personally like to picture atoms vibrating. The higher the temp the faster or stronger they vibrate. At 0K which is the lowest possible temperature, the atoms are completely still (aka no “vibration” they may still move in space).
Now back to your original question, how does an AC work? ACs, just like fridges use heat pumps to “move heat”. They basically pump a liquid that’s cooler than your room into the AC unit where the air and the liquid exchange heat energy. This cools down the air and heats up the liquid. Then the room temped liquid is pumped out of the room and usually outside the house where it is cooled down again. You may now wonder how this works when the temperature outside is higher than inside. To achieve this ACs use some black magic, also known as thermodynamics. Pressure and temperature are interlocked with each other. When you manually increase the pressure in a room, that will lead to a higher temperature. ACs and other heat pump based systems are able to utilize that phenomenon.
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