So, even stranger yet, it actually only doesn't work on certain pages. I created new pages and tested them there, and they all worked except one.
The one that didn't work finally did work when I no longer included a textbox which was validating URL and no longer submitted a website starting with "http://".
So I've got my forms working again. But I really don't understand why they didn't work before... And I kind of wish that I could still use the old page slugs rather than the new ones, but these will work...