So after trying several things independently and none of them working, I ended up taking the nuclear approach. I deleted my .local and .config folders, clear the cache, then completely uninstalled Plasma (I had GNOME installed as a backup). I then rebooted and re-installed Plasma. Everything is working fine now.
Edit: Ok, I found the “real” solution. I set the environmental variable “QT_QPA_PLATFORMTHEME=kde” in /etc/environment. For some reason, this screws up the wallpapers and Plasma desktop settings on my laptop, but not on my tablet.