I guess you know that replacing the linen is always an option.
The wax paper trick works but then the linen will feel stickey.
Some people hate that worse than the fuzzies!
What I would do is first clean the linen then give it a light coat of sanding sealer to sink in and firm it up.
A light sanding with 600 grit will then get rid of the fuzz and bring back the feel of smooth linen.
If you sand the linen without some form of sealer you might fuzz it up even worse.
You can protect the finish near the wrap with masking tape.