Drywall – Applying joint compound over old and dirty drywall

I know this isn't the answer you want to hear, but I don't think priming and painting will hide the tape.

Before you apply any more primer, I'd add another thicker coat of mud. Slather on the mud thick first - it doesn't have to be smooth. Then smooth it out by angling your 12" knife so it's almost parallel with the drywall as you drag it along the surface. You'll have to press fairly firmly in order to avoid air gaps appearing in the thick coat you're smoothing out.

You may have to get a feel for it. If you push too hard and can see the tape again, redo that section until you get it just right.

Wait for the thicker coat to dry, then sand the edges as smooth as you can get them. Very carefully/lightly sand over area covering the tape if you didn't get it quite smooth. You may need to do a second coat along both sides of the thicker coat to make it join the drywall surface smoothly.

Though you might be able to mud over them, you're looking at 1-2' of mud (feathered) around each screw to make it not noticeable, which basically means you'd be covering the entire piece of drywall with mud. Like pretty much everything in construction, it will be much easier to do it properly now rather than try to fix it later.

If your driver is slipping out, you are probably not using the right bits (or your bit is worn out) or are not pushing hard enough on the drill to prevent the bit from slipping out. Make sure the bit fits well in the screw head: it should not have any room to wiggle in any direction, and should not slide around. #2 Philips is the usual here, but depending on the screw head it may need to be a sharper/longer or blunt tip.

It could also be that your screws are not the right type. If you're going into wood studs, generally you should use coarse thread screws. If you're going into metal studs, it should be fine-threaded screws.

Since the board is also 5/8, it may be that the screws are not long enough. Try longer screws, which should get a better grip on the studs behind so once they start going in, they'll go in as deep as you need.