Ok, so I am answering my own question, denatured alcohol. I found after some searching.

https://www.wisconsinhistory.org/Records/Article/CS4249

I still need to figure out how to treat the dark stains , I'm afraid that they are in the wood. Bleach dabbed on with q-tip. I have no clue.

Depends on your finish if alcohol will smooth it out, or not. If it's shellac, it will work, if not...well....
Try a mix of 2/3 alcohol and 1/3 lacquer thinner...same method.
As for the black water stain....if it's into the wood, about the only think that really works is sanding it out. You can try removing the finish in that spot and bleaching....but I'll bet the stain is deep. One thing you can do, is paint. Yeah I know!....but mix up some artist's oils in the color that will match the rest of the wood, and cover the stain...have some paint that is a bit lighter and some darker....and try to make it follow the grain of the wood. Then after bout 3 days drying, put a clear finish on top...oil based not water.
I have a 150 year old victorian piece that my mother used for a fish tank stand...with the inevitable results!....and I made it cosmetically fine.

Table is definitely natural shellac. I had worked on it a few years ago with denatured alcohol to smooth out the finish. I guess I just need to put the pedal to the metal and just try. Thank you for your suggestions. I am 100% sure black stain from water is in wood. I had table covered with blanket and dum-dum hubby put a cooler that leaked on it. I removed cooler probably a week after he sat it on table and discovered stain. I kind of went ballistic. It was my great grandmas table she got as a wedding present.