No primer, no paint, no sanding except for areas that obviously need it.
I've done this probably a dozen times. Drywall mud is best. Personally I use Hamilton's Taping Mud (drys harder) to skim coat or float the textured walls. And each time I used it over a previously painted surface, followed by spray or hand texture.
In fact I learned a trick a long time ago when prepping to spray texture over smooth walls such as a bath that's been painted with a gloss or semi-gloss, and that is to skim coat the wall using the drywall compound first. That way the texure won't run or sag as it might if just applied over a smooth glossy finished wall.