First, I would make an earnest effort to acquire black screws.
Failing that...
I use WalMart's brand of spray paint. It dries in about five minutes. Just be sure the screws are clean, and free from "mill oil." Stick them in a piece of cardboard, or styrofoam and spray. The longer (read days) that you let them dry, the harder the paint gets
+1 (Except I have more experience with Rose's One Coat spray paint)
Also, a lot depends on the original finish on the screws. I would avoid using zinc coated altogether. It's just going to flake off over time anyway.
Ideally, I would sandblast them to remove the old finish and give the surface tooth, but sanding with fine grit sandpaper will work. It's just harder to get into the slots that way.
Several thin coats is MUCH better than trying to do a thick coat. You can speed of the curing (hardening) of the paint a bit by warming the metal before application. Ideally, setting them out in direct sunlight on a hot day seems to work best. You can use a torch, but it's very easy to overheat them so the paint burns off as you apply it. And you really cannot reheat for additional coats that way.
Once the screws are driven in place then You could touch them up with Testor's model paint and a fine brush. Dont bother with the cheap brushes they keep by the model supplies. Get a quality brush from an art supply place. It makes the job SO MUCH EASIER.
Why not do the whole job with the Testor's? Because it is much easier to get a smooth finish with spray paint, and the touch ups won't stand out so much.