Are you logging in with your library card/pin or with an overdrive account?
I’d suggest you try the overdrive account (it’s an option when you’re going through the process of connecting the library to your device).
See this post where I go on a bit of a rant but also provide step by step instructions to use multiple libraries on one Kobo. Which would also benefit you since you’ve got multiple libraries from what you’ve said
https://www.mobileread.com/forums/sh...39&postcount=6
Note the above is for US libraries but I presume Overdrive used the same infrastructure for all accounts. And while your library might not use a card number/pin I’m guessing they’ll have a library specific account (this is what you’d want to link to overdrive but not enter on your Kobo).