Why do I get auth error concerning servers on my iPad after iOS 7 update using Overdrive Console?

Worth trying to install the Bluefire app and authorizing that.

It is possible that Bluefire will do what you need instead of Overdrive,

also possible that authorizing Bluefire will allow Overdrive to work, as you really authorize the device, not the app.

(Unfortunately, it is also quite possible Bluefire won't authorize either ... but worth a try).