Thanks for the reply. Looking into it more just now, I see what was causing my confusion. I just re-tried editing some buttons that I had added while developing... removed the link, and re-added to the page using SS's chooser... and it inserted the relative link, but kept the demo URL when hovering over those buttons (in the editor).
However, when loading the actual site with its public URL via a private window, those links DO show the correct URL.
I was getting worried when I saw the links on SS's backend weren't updating, when that's not what I should've been looking at!
Thanks again for helping me figure that out 🙂