Hi all! I'm a newbie here and I have the exact same problem with my PRS-650!
I authenticated my computer with the Adobe Digital Editions, but it would not recognize my attached ereader at all (I could open files on it if I wanted to, but wouldn't do the automatic authentication that Adobe's website said would happen at all). I've also done jashsu's suggested steps on my own with the exception of step 3 (because Sony's software didn't ask if I wanted to associate an Adobe ID to my ereader) and ADE still will not acknowledge my PRS-650 (not unexpected with the failure of step 3).
The Sony software sees the ereader fine and once the computer is authenticated, it can open my DRM'ed book and also copy it onto my ereader, which will be able to read it is well. But there's a big catch: if my ereader is fully turned off or I open something else (in the case that just happened, it was a pdf from an SD card) and... BLAM! The file is blocked. The only thing that I've tried which seems to work is to delete the copy on the ereader and copy the file over again. Very irritating and I can imagine getting very annoyed when this happens where I can't just fire up my computer to recopy the file.
I have a bunch of non-drm'ed files and had used Calibre a few times with my ereader. Could this be a problem? Has anybody had this work and were there any tricks? Maybe I need to reformat the ereader for a clean slate and try again? I'd rather not do that if I don't have to...