One solution (I have seen work) is to squirt 'Great Stuff' into a foot or so of
the open end, and let it form around a skewer. After it dries, pull the skewer
out for the drip hole.

Great Stuff (comes in a can) is a foam insulation, much like styrofoam, used to
seal window joints, and so on. It is transparent to RF below about 3 GHz

When you squirt it in it hardens up and suppress the driven oscillations.Bummer
that you have to pull down the antenna though...

Also, some folks use nylon rope in the elements as a damper. Perhaps someone
who has succeeded with that will respond.

Best of luck on this or other solutions.

73,
Chip N1IR