I've been playing around with this a little bit and it looks like as Pjama suggested, they changed their security a little bit. I tried generating a new auth token in the app via the clone code and that didn't work for me, but I did the same in my browser and copied its auth token into this plugin's config and that worked perfectly.
No promises on timeline since I have limited time to work on this, but I'm going to see if I can work the newer token generation method into the plugin. Alternatively if anyone is interested in working on it I'm happy to share my findings so far.
If anyone needs to get their plugin working, you can follow the steps below. I don't know how long this lasts for, so you may end up needing to redo it after some hours/days. Also the instructions are based on Firefox, hopefully they apply to other browsers as well but if not just ask and someone will try to help out.
1. In your browser, open your dev console (ctrl-shift-I or F12 for me).
2. Go to the network tab, click the cog at the top-right and make sure "Persist Logs" is on.
3. Go to
https://libbyapp.com/.
4. In the dev console, type "sentry" into the filter bar.
5. Click on the furthest-down entry that is a "GET" request.
6. In the right panel, go into the "Request Headers" and find "Authorization" -- its format should be "Bearer" followed by a long string of characters. Copy just the long string.
7. Make sure Calibre is closed, and open the plugin config file (%APPDATA%\calibre\plugins\overdrive_libby.json on Windows, unsure Mac/Linux paths).
8. Find the line for "libby_token", and replace its value with the one you copied in step 6.
9. Open Calibre and the plugin should work as expected now.
Hopefully this is helpful to people.