I saw a few topics on this that didn't address my issue, and since I found a solution, I figured I'd post it. The issue is that when using a DS4, the controller seems to constantly input up-left, most noticeably resulting in uncontrollable scrolling on the main menu, but also in constant movement in that direction in gameplay as well.

The fix I discovered was to open the config.exe, located in the game's install directory (for me, that's C:\Program Files (x86)\GOG Galaxy\Games\Ys Origin ), and navigate to the Input Settings tab, then move the deadzone all the way to the left. This centered the crosshair which was, as you might guess, previously stuck in the top-left corner of the box.

Hopefully this helps someone.