Never mind. It turned out to be an issue with WP Super Cache. For some reason, I guess certain cached items weren’t clearing.
Ok, I found my own problem here. It was staring me dead in the face. Somehow these two lines had been removed from my htaccess file:
RewriteEngine On
RewriteBase /
I think it was caused by a plugin, but I don’t know why it didn’t cause an issue until now.