Although you get some quality loss when re-recording, it is not that important for audio books.

Loslless DRM removal software works faster and it is free, but usually it doesn't live too long.

I'm more aware about Windows Media DRM, but if I'm not mistaken you are usually not required to update iTunes, so you may use older versions of it with lossless DRM removal software.
p.s. I don't really know whether Overdrive offers iPod files or you get files for your iTunes that can be copied to several iPods later on.