My local library in the UK uses Overdrive as does Mills & Boon (which supplies my wife's monthly fix of Romance books). Both mistakenly say that Adobe Digital Editions works with the PRS-T1. It doesn't. A bit of googling will show you the problems that others have had. In fact, somewhere on the Sony site, they admit that it doesn't work with Adobe Digital Editions. One way of solving it is to use the Sony software which means more unwanted software clogging up your PC.
I dunno, I can get Overdrive to work on my PRS-T1 in the U.S., but only by navigating to my library's Overdrive site using the Web Browser instead of the Library button. Have you tried doing that?

I can also use Adobe Digital Editions, but it needs to be set up to the same account as your reader.

That is, load the Sony software and enable DRM using your Adobe account. Then go to Adobe Digital Editions and register the DRM using the same account.

You should then be able to pass files back and forth between Adobe and your reader.