I had endless problems with the ADE2.0/Overdrive/Sony PRS-350, and I think it is the same with the 650.
Having the Sony plugged in stopped ADE doing anything with Overdrive books.
(1) Use the older and much more reliable ADE1.7.2.
That fixes lots of issues including my Sony/Overdrive issue and probably yours.
(2) Use ADE2.0 to borrow the books and download the DRM .epub file.
Also to authorize the Sony.
Copy the .epub file to the Sony using a file manager such as Windows Explorer or Finder.
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