Try using the Bluefire app. Quite a few people seem to have problems with Overdrive.
It is even possible that once Bluefire is installed and authorized, you will find Overdrive starts to work properly.
The only thing you can do with Overdrive but can't do with Bluefire is to borrow and download library books directly on the Android;
you need to use ADE on a computer to download the .acsm and turn them to .epub, then copy the .epub to the Android.