Ok so first off, look up all posts a stickys by the Junkman, he will go over every nuance in detail of detailing.
On to your query.
If the paint feels rough to the touch you likely have contaminants on the paint causing this to occur, even after the car is washed, they remain on the paint causing the rough surface.
Dawn wash will not hurt anything, as claying your car will remove most of the protection you had on their regardless if not every last drop. Clay is the step you need to take to remove this rough feeling. Wash the car thoroughly, then get a MILD clay bar which the one you have likely is. Go watch any of 100 youtube videos on how to clay a car properly.
After claying, you will need to ideally, Polish the car to restore the shine and luster and remove the marring caused by the clay bar. Likely with our soft clear, you will have swirls all over the car once done with a clay bar.
Now you polish the car up to a deep deep shine, then protect all of your hard work with a paint sealant or a nice quality wax or synthetic substitute.
Mix brands all you want, they all use the same stuff anyway with different scents!
Be aware, if you DO NOT polish after the clay, you likely wont be satisfied if you are concerned with paint condition. Waxing after clay simply locks in and protects all the micro marring scratches you just induced with the clay bar!
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