The single word would replace "pinching, fanning and shaking out", perhaps as:
He ________ the pages, and the note or the stamp or the pen fell out.
"Riffled" misses, imo, as it suggests the pages of a book, though the process of grasping a book by its binding and shaking... heck, I don't know a word for that either!
The word (i.e. ______) could be done with an untidy stack as long as one could pinch the stack sufficiently