Dear Jonathan,  Greetings from The Bronx,  Perhaps your problem is
not that the music books are sticking, but that the rubber pinch
rollers are not doing their proper job.

- How old are those rollers, and have they ever been replaced (or
recovered with new rubber)?  With age they may have become hard and
less able to grip the cardboard.

- Is there an adjustment to put more pressure on the rollers so that
they can grip the books tighter?

- If this is a keyed system, the keys may be adding additional drag
to the books, and reducing each individual key pressure may help to
alleviate the problem.

Hope this helps,

Sincerely,
Dale F. Rowe
Bronx, New York, USA