Meguiar's Ultimate Compound follow by Ultimate Polish using a buffer/DA should take the swirls out..unless they are severe. I would also walk through your wash process and figure out what it is causing the swirls on your paint (unless they were there when you got the car?). There's no point in paint correction if your washing is going to make swirls again.
As far as washing a car using hard water..it is possible. It is bit more time sensitive though. The main thing is do not let the water DRY on your car. You don't necessarily need soft water..just be sure to wipe the car down when you are done washing it. If you are washing the car in the sun or hot temperatures (which I don't recommend anyway), you may want to wash the car one panel at a time, drying it as you go.
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