It was at first {Page \* MergeFormat} and then after I reinserted with Alt + Shift + P it was just {Page}.

After doing so, the page numbers would still change after hitting Ctrl + P (and the {Page} was lost in the affected sections. If someone posts about this in the future and it seems as though the poster has manually typed in page numbers, that might not be the case. I saw some old threads here and on other forums, where the response was that the poster was probably typing in page numbers. This bug or interaction or whatever we might call it actually removes the {Page} from the page numbers of affected sections when the user tries to print, and all the page numbers in the affected sections then appear as manually entered numbers (usually the same number throughout the whole section).

Something with the cross-references/bookmarks was interfering and appears to have been an issue in at least one other case (linked above). I didn't have time to run trial and error to single out which ones, so I ended up removing all the clickable cross-references to the table titles and figure captions (e.g., see Table 4.3, where 4.3 was a cross-reference to the actual table title of the same number). Then I was able to print the document without the problem described above.

I wonder if one of my cross-references was somehow connected to a footer area, although I don't recall that being the case.

Not a perfect solution given that I lost the cross references, but it had to be done with the timeline at hand.