All of my saxophones have "barrel shaped" adjusters which slide back and forth in tracks located at high F, G#, and low C#. I don't have the slightest idea how or why to use them and have never seen any instructions on this, yet all horns have them. Can anybody explain these to me?

Those are not "adjustment screws" as the term is typically used in saxophone repair- adjustment screws usually meaning the screws on the bar key in the upper and lower stacks to control the relationship between the stack keys and the C# (upper stack) and F# (lower stack).

They are however screws that can be used to make a very handy adjustment! They control the distance the force-transmitting arm is from the fulcrum of the key barrel, thereby allowing you to change the relationship between the actuating arm and the key so that more or less travel in the keys can be allowed relative to the desired key opening height.

In other words, you can slide the arm in the track to change how much travel in the finger means how much travel in the pad cup.

This is helpful for making keys work well together without lost motion while allowing for variance in key heights.

sometimes, you have to say "I always wondered about that".....thanks, Matt!

I have seen some of these plastic covered, others bare brass. Is there an advantage of one over the other?

Nope, just different ways to skin a cat. The adjustment material there needs to be slick, smooth, quiet, and not take an impression over time. Unfortunately most often you see cork or natural felt in there- although natural felt isn't as bad as cork. I use teflon-coated synthetic felt, which doesn't care if it is sliding across smooth brass or smooth plastic.

Is there some reason you don't just glue the Teflon to the metal bar?

Teflon against adjustment material is not as slick as teflon-covered adjustment material against metal.