Most people forget that dirt doesn't come off easily unless it's scrubbed/wipped off using a mild detergent or a lot of pressure with a harsh detergent. All touchless washing systems (including your dishwasher at home) uses harsh detergents to get the dirt off. Those harsh detergents generally get everything off including the stuff underneath the dirt like the wax and paint... Just look at your own glassware or dinner plates after a few times in the dishwasher
As H15A stated, get some good quality washing cloths like microfiber.
If you want a good detailed guide on detailing (sorry couldn't help myself), check out the autogeek website - Auto Detailing Facts, auto detailing Tips, How to detailing Guides, how to polish, how to wax, DIY detailing, do it yourself guides. Please note that they sell stuff so many of the products that they use are "advertised" product placements but the general process is correct and the warnings are pretty good. They might be a little OCD however