First, I would like to preface my comments by stating that working
on antique musical boxes is not something for the amateur nor is it
something to be taken lightly.  They are very delicate mechanisms and
a simple mistake can easily destroy the cylinder and musical comb
making the box irreparable.

Reading the description of the governor it sounds like it almost
certainly has a bushing issue.  If you apply pressure to the gears and
they move on their axes rather than in rotation, then the bushings are
shot and need to be replaced.

The good news is that, if you are not equipped to handle this, a clock
repair man should have the tools and replacement bushings.  Note that
this person is not a "shopping mall watch repairman" today, as all they
are capable of doing is replacing a battery, but an actual repairman
that will fix a clock.

Elijah Shahar