Unfortunately, the only way I know of to fix those covers is to remove the DRM and use Calibre to add them. It's a bit of work, and very annoying. I'm not sure why B&N's reader can't read the covers if it can read the content, other than B&N just not wanting to fully support non-B&N content.
If half of the books appear without a cover in ADE, then the chances are the books came with no cover.
My DRM free books appear with their covers in ADE. Anyway, I installed the Kobo desktop reader as you suggested. While the covers appear in my library there are not any when I choose to open a particular book. So there mustn't be covers attached in the files I got.
I guess I have to accept that most, if not all, of the ebooks I' ll buy will have no cover but just a title page.
Thank you both