If any one has found a solution I would be grateful to know about it. I uninstalled and reinstalled as was suggested. I am registered at Adobe and Overdrive welcomed me! But I cannot "Authorize" my iPad.
Unfortunately, Overdrive has stated they are having issues with their newly upgraded to iOS 7 app:
Their only suggestion is to delete the app and re-install it. However, it will cause you to lose everything and have to sign back in again, but that's all they have right now. I would assume they are feverishly working on a fix, so, if you don't want to go through the hassle, maybe you can just hold off using it until they update/fix it.
My Overdrive is working normally since the update, I didn't have to do anything
Editing to add, the overdrive app itself updated a couple of weeks before the Ipad update, I did have to sign in again with the adobe ID, not the Apple ID