I don’t know if this applies, but aside from eliminating any play in the system, is the rudder wiggling because it is no longer symmetrical as the water flows past both sides?

Our boat has a little bit of play if you jiggle the tiller while sailing upwind, but our rudder would not wiggle on its own while sailing.

(This is our first year with a J/30, so I don’t know the boat inside and out)