Thank you for the suggestions. When I removed my init.el file, AUCTeX's mode loaded fine so I knew it must be a conflict somewhere else in my init.el.
I had the following lines in my init.el to try to decrease my emacs startup time (as emacs had been slow to load):
(setq package-enable-at-startup nil)
(package-initialize t)
(let ((default-directory "~/.emacs.d/elpa"))
(normal-top-level-add-subdirs-to-load-path))
When I get rid of the first line, AUCTeX loads fine, so this fixes my problem. (It turns out that first line really wasn't helping emacs start that much quicker anyway).
Still, this workaround seem a little odd to me. First, It seems odd that I can't start AUCTeX "when I want", i.e., by using the line (load "auctex.el" nil t t), even though this worked for me in previous versions of auctex (11.89 and 11.88). Second, it also seems a little odd that I can't do something like (require 'auctex) or (use-package auctex), even though I can load all my other 30+ packages that way. The concept of having nothing in my .emacs and the package still being expected to work definitely confused me at the beginning.
Regardless, this fixes my problem. Thank you very much for the help.