Arrrrrrrg.... I See Spots

Hi. I was cleaning up a filthy hall mirror I purchased from a barn sale when I noticed these spots a couple of hours later. I'm using mineral spirits to clean the dirt, coal tar, wax off. I used a pad of 000 steel wool and flooded the surface with the spirts, wiped off the resulting gunk and let it air dry. When I came back a few hours later, these spots were visible. The finish was uniformly dark before cleaning.
I tested the varnish before starting the cleaning and it tested as shellac. The panel is cherry wood veneer over secondary wood. Did the shellac absorb moisture from the air and turn white or did the shellac 'bunch' up?
Then there's the question of how to 'fix' the spots. Should I try some alcohol stain to see if the finish comes back uniform? Any help is appreciated, thanks!

 

Joe, you have started to remove the shellac. Now you need to finish the job. You can remove some or all of the existing shellac finish by rubbing it down with 0000 steel wool and denatured alcohol. Once the wood returns to its clear/ non white wood tone, you will need to re-shellac the wood. The hard part is going to be if the shellac was tinted. If it was you can just remove all of the shellac and re-apply shellac or switch to polyurethane. But wait to see how others recommend repairing you piece.

 

I am not sure if denatured alcohol will do the trick or not. If not, try lacquer thinner. Wet a rag and wipe quickly or you will remove too much finish. Apply multiple times if one swipe is not enough. Let dry completely before each attempt. You may need to apply a new coat of finish once the spots are gone.

 

I'll try the lacquer thinner and report results. Thanks!

 

The lacquer thinner did the trick! The varnish which was not picked up by wiping the surface cleared up the spots. When the weather turns a little dryer and warmer I'll apply a new coat. Thanks for your help!