Some of the dish soaps have dispersants that actually promote water spots, that is the primary reason they don't recommend them for car washing. Water spots can be removed with a lot of effort using a clay bar, then re-polishing your paint using whatever products you prefer.
For me, my process is:
-wash car (using auto car wash soap)
-bug and tar remover
-wash again (same soap)
-wax remover (from an auto parts store)
-wash again (same soap)
-clay bar on cloudy areas or water spots
-3M imperial finishing compound on areas that require either orange peel removal, or spots that clay bar couldn't fix (BE VERY, VERY CAREFUL!!!)
-wash again
-3M Hand Glaze
-wash again (by the time you dry the car, you'll need sunglasses)
-apply the best pure carnuba wax you prefer and ensure you don't leave hazing or swirl marks or wax residue in recessed areas (I use masking tape to keep wax from going there in the first place)
From there you can use a detail spray and regular washes and your wax should last an entire season.
For some cars, it's taken me 14-16 hours to get them up to the level that shows the paint correctly.
Regards,
Aaron