I've check the forum and I cannot find that this has already been addressed. If it has, please point me to it.

In Myst III, anytime I click and drag, in game or in the menu, thick black bars start cycling up the screen. It reminds me of an old VHS that got misaligned, if anyone can remember those. This issue did not happen in either Myst or Riven with the same settings. I've attempted to run different compatibility modes, uninstall and reinstall, etc. but nothing seems to be working.

This is not a game breaking issue, but it is annoying.

I've just finished the game and never experienced anything like that.
But i did used newer version of ResidualVM than the GOG's is using for this game, which may fix this issue for you, it provides better stability and even some fixes for this game and few others, also allows you to play in nice fullscreen-widescreen mode(no black bars).

https://www.residualvm.org/ - just download latest version, install it, then launch it, and you should see Add Game button, so just click it, navigate it to your myst 3 folder and you should be good to go.
You can also play with some settings in Residual (I've used OpenGL and then ofcourse that widescreen mode, with fullscreen and Keep Aspect Ration unchecked). Worked like a charm.