Hello,  Well, I guess I can't resist adding my 2 cents worth to the
refinishing question.  If you suspect that the finish is wax, you can
wash off the wax and the dirt with paint thinner.  The thinner will not
dissolve the shellac or any varnish.

If the finish is indeed shellac, you can wash off the top layer with
alcohol and a lot of paper towels.  Start at the top and work your way
down.  If it's just the top that needs attention, mask off lower parts
so the dribbles do not cause a problem.

The first layers will make the paper towels brown or black.  When you
get only gold, shellac colored residue, you're down to the original
finish.  Refinish with new shellac and you're good to go.  Or you can
wash off the shellac down to the wood and start from scratch.  If the
shellac if very thick, you can soften it with alcohol and then scrape
it off with a cabinet scraper or a sharp putty knife.

Go buy a gallon of shellac thinner at Home Depot.  That's a lot of
evaporation so please do it outside if possible so your eyes don't
fall out.

Regards,
Craig Smith