Is there an issue with organizing bookshelves on Shabbat? I could see this potentially being an issue of borer but I don’t know.
No it is not possible to organize your bookshelf on Shabbat because of the issue of borer which you mention.
Peninei Halacha writes
Just as the prohibition of Borer applies to food, it applies to other items as well, such as books, [...] Similarly, one may not sort a mix, and all the more so one may not remove the items he does not want.
Seforim may not be sorted and ordered on Shabbos. According to many poskim, the essential prohibition of borer is a prohibition of sorting, and the condition of “waste from food” is only intended to reveal that the act is an act of sorting and not an act of eating. Therefore, seforim should not be sorted on Shabbos. See Tosafos, Shabbos 74a; Biur Halachah 319:3; Shulchan Shlomo 319:4.
R Dovid Ribiat (The 39 melachos, vol II, p. 444) also writes
Seforim in a shelf are considered "mixed" and may not be rearranged into their correct order.
On the other side you can pull out and return books to bookshelves in the usual way (Peninei Halacha)
Books on a shelf are not considered mixed. Therefore, one may remove books he wishes to learn from, even if he will not actually use the books until hours later. One who used many books may return them to the bookcase on Shabbat for the sake of neatness. In the course of returning them, he may put each book in its proper place (R. Shlomo Zalman Auerbach in SSK 3, nn. 220 and 239).