how to "sharpen" "free up" 20yr OLD Berol China Marker (Grease Pencil)?

how to "sharpen" "free up" 20yr OLD Berol China Marker (Grease Pencil)?
Yeah, I pull back the string enough to pull off the next layer of paper, and that works until maybe 5 inches of paper get unwound, and then (not again!) the paper breaks off at the surface.
What I've done for years is to apply a sharp serrated knife four or five places at the front, leaving five or six (the n + 1 trick!) rectangles of paper STRONGLY stuck to the grease pencil's "lead" (for lack of better name), then, with much difficulty, try to pry up enough of the paper rectangles to be able to write with the thing (for a short while).'
Besides "throw away the damn things and buy some new ones", I've had pretty much the same problem even when they were new.
Down in the shop I have a Dremel set. Would that help? And just how would I use it on the grease pencil?
Thank you!
David
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