I could not get Overdrive working on my Kobo: no matter what I did, it would not let me add a library card to the system. The Libby app (whether an actual app on my phone or the web interface) was working perfectly fine with two different library cards.
After much searching, especially on `mobilread.com`, I found a solution. I logged out of my account on the Kobo. I then logged in again and was then able to add a library card to Overdrive. This did mean that all Kobo files had to be re-synced. However, as all of my books are managed by calibre, I wasn't worried about losing any of the actual (and specifically any side-loaded) contents.
I can now borrow and read library books straight from my Kobo again or borrow them via Libby and then sync to the Kobo. Now, if only koreader supported Overdrive...
This was all motivated by my wanting to read the Dune graphic novel which I could only seem to be able to read via a web browser (). I've yet to see how readable this will be on a grey scale Kobo, mind you. Watch this space. I might have to go out and buy a colour one...