Bo, there are nothing awkward about those screws.
For instance, let's see what happens to the mechanism when a screw is in action. Take for example the A keys (first octave). There are two keys responsible for doing this note, but you only have one finger available for that. So the screw is responsible for regulating the amount the "second" key will go down (or not so down) to make a perfect closing of both.
If you turn the screw a little to the right, it will go a bit down and will cause the second key to go further a bit down. At the contrary, if you turn it a little to the left (counter clockwise), it will raise the screw a bit and the support pad of the second key will be pressed a bit less, causing the second key to go down a bit less. The commitment here is that the screw is in a position so that both the primary (where your finger goes) and the second (the one the goes down "automatically") keys make a perfect match when closed, avoiding any leaks of air that produce the "different" sound you noticed in your previous experience with changing the screws positions.
BTW, all screws that might exist in a flute have this very same function (except for those that hold the rods in place...). There are flutes that simply don't have ANY screw. You might be thinking "what to do when a flute like this stop working?". That's why we have good technicians...
It's just to help you get rid of this sensation of doing something very wrong, which it's not. You just need to learn how to use it in your favor and when it's really needed. Sometimes it can help you spare some money.