There is a split plastic insert that is squeezed between the mounting post and the sunvisor sleeve when you tighten the screw. It has likely worn through or shifted such that it no longer makes contact to cause the necessary friction to keep the sunvisor from moving on its own.
Loosen the screw enough to slide the sunvisor off the mounting post and look at the insert. It is white plastic, is about 3/4" long, and has a flange lip on one end to keep it from being shoved all the way inside the sunvisor when being installed. If it is missing or damaged beyond repair you can get replacements from a junkyard (most GM sunvisors have a similar insert) or wrap some electrical tape around the post to act as a surrogate insert. You may even find replacements in the HELP packages in auto supply stores.
I believe Theoger has a writeup about sunvisors in his cave.
Electrathon,
Do you have any pictures of the LeBaron sunvisors that you'd like to share? How does the color match with the Fiero interiors?
Good luck,
Carroll