Spot prime the damaged drywall with gardz. Then float the walls. Trying to sand texture smooth is a mistake. Just float new mud over it and sand that down. For $20 at Lowes, you can get a hand sander you can attach drywall screens to a shop vac to suck the dust as you go. I do this on a regular basis with textured walls before I hang wallpaper. With a good bag in the shop vac its virtually dust free. I use lightweight joint compound, and 220 grit screens, sanded at an angle, or else you will see the screen marks in the wall. Once the wall is sanded and smooth, gardz the whole thing, to make it bullet proof. There will still be some screen marks, so repriming with something thicker like 123 is a good idea on top of the gardz.
You dont HAVE to gardz the skim coat, but I like to, because it penetrates the surface and makes it bullet proof. 123 will work as well, as long as you get the dust off the wall. With my shopvac sander, I dont worry about it, though, but wall dust is much less of a concern with Gardz, as it will straight up penetrate a little into the wall.
Wiping the wall down with damp rags will help with the sanding marks though. Generally, three rolled coats will cover the marks though, but you want to diagonally sand. OR, if you arent concerned with dust, you can sand again with actual sandpaper, and it wont leave the marks.