Any 70's home I've worked on used nails not screws.
One we own there was an earth quake near by and it popped every single nail.
They had used 1" smooth shank, small headed nails. (looked like box nails)
Adding more screws is the way to go.
What you may see happen as the drywall gets pulled tight to the stud the area where the old screw or nail is may pop out or bulge.
Just give it a tap with a hammer and remove any loose material.
I use a 4" wide knife and often use the outside corner of the knife to push some of the mud into the recessed area so there's no air bubble trapped that will cause a low or loose area.
The only trick is to use thin coats, so very little if any mud is on the wall it's self.
Once dry a quick light sanding, go back and check every one of the repaired spots looking for any low or high area, rub your hand over them, if you can feel a flaw it's going to show when painted.
Wipe down the walls with a damp rag to get rid of any sanding dust, paints not going to stick to dust.
Once it's primed look it over again and fix any flaws before painting, primer makes the flaws stick out, and no paint is going to fix mistakes.
Use a sanding pad to sand so it stays flat on the wall, sand in a ciruler motion so you do not get lines.
I run my wet dry vac with a drywall bag over the filter holding the hose below where I'm sanding for less clean up.
Opening up a window, removing the screen and setting a cheap dollar store box fan in the window blowing out also helps with the dust.
