I figured it out.
For one, I had notification turned off in Retroarch, and I had save state thumbnails off, which was throwing me off a little.
I pressed F2 on my keyboard and then navigated to Retroarch\saves\the name of the core I'm using, and I saw the save states there, so I knew it was actually saving.
Now that I had notifications on, when I tried to load a save state by opening the quick menu in Retroarch (F1)/save state/load state, I saw a notification appear that said, "You must pause or disable Achievements Hardcode Mode to load states".
I went to Retroarch settings > Achievements, and turned hardcore mode off. I can now save and load games from the pause theme in Launchbox.