There are bugs with ADE2.0 that mean the combination ADE2.0/Sony reader/Overdrive library book does not work correctly.
This is one of the vast number of bugs in ADE2.0.
There are lots of bugs in ADE2.0. Try replacing ADE2.0 with the older but more reliable v1.7.2.
(You can have them both installed at once if you like.)
Version 1.7.2, it is a little difficult to find, available on Adobe site for Windows and for Mac.
The forum software is sometimes corrupting the link above. There shouldn't be a blank in 'editio ns.html'. The following redirects to the same page: http://tinyurl.com/diged172
To use ADE (either version) with a Sony device you need to install Sony Reader.
Download from https://ebookstore.sony.com/download/
Even if you don't use Sony Reader, it installs a device driver that enables ADE to interact with the Sony.
(Not sure why Sony makes this awkwardness that other devices don't have, it slightly marrs a really excellent lightweight ereader.)