You are aware that you can control the temperature that the AC targets? I live in the US and so have AC and I set it around 29C under normal conditions (if it is unusually humid I find I have to set it a bit lower to be comfortable.)

How does it make you feel sick? I don't understand how that could work physiologically.

Same mechanism why you get more sick in the winter. Periodic thermal shocks and therefore periodic capillaries expansions and retractions in you airways, sinuses. People are quite susceptible to it here in central EU, me included. US people seem to be kinda conditioned to the constant thermal shocks.

Here many only have and use AC in their cars and only in peak summer and I remember there used to be warnings in TV news that you should turn off or start gradually decreasing AC intensity before arriving to your destination a getting of your car. (It was usually as a part of some interview with some GPs, discussing why are people getting more sick in peak summer.)

That's not what causes people to get sick in the winter. Warnings about AC use is car are nonsense. Why would you believe that?