Hello sammy,my advise assumes you have basic knowledge of refinishing. Otherwise I advise hireing a pro or at least an accomplished hobbiest.
It is absoluty nessary to determine if silicon is present on the finish. Removing/sealing it is a job within it self,and failue to get silicon free surface will twart any efforts. Look carfully at all products you have used while cleaning to see if they contain silicon. No? Good! That doesn't mean previous owners didn;t put silicon on it but that's another story. As insurance against unknown contaminates,use a pre-cleaner from a paint store before doing anything else.
Your statments about extreme sensitivity to mousture leads me to believe the existing finish is shellac. I farthure believe it was "french polished" using shellac. While shellac is a great seal coat under most other than water base topcoats,it should be thin as possible.
What I perpose is scarry but that's where experience and sneeking up on it prevents disaster. Denatured alcohol will desolve shellac where it can strip,blister or damage lacquer,the effect is very different. The inverse is true of lacquer and medium thinner. My plan would involve playing around to determine which is on there now. If it confirms to be shellac,I would remove the finish by wipeing it away with denatured alcohol,sanding,sealing with a single coat of 1# shellac then top coating with lacuqer. You must honor all dry/cure times between coats. I like lacquer because repairing latter damage is simple and any degree of gloss can be easily attained at time of application and years down the road as well. Shellac shares some of that quility but as you discovered,it will blush when exposed to moisture.

klem1 has some good advise. I can elaborate.

First, what you want to determine is silicone (note the "e") oil on the finish. The most common source of this is Pledge furniture polish. Once you have silicone oil contamination, it's nearly impossible to get rid of it completely. It will cause many finishes to "fish-eye," cratering of finish when it's wet because the silicone oil destroys surface tension. I don't know of a reliable test for this, but I can usually spot it by the sheen or by the way finishes or stripper react when applied. Even if Pledge has not been used on the piece for decades, the oil is still there. If you can spray shellac, that's one way to seal it in. But otherwise, you might need to have it professionally stripped and refinished, as the pros know how to deal with Pledge contamination.

The finish ion there is most likely shellac or lacquer, unless the prior owner or someone else did a DIY finish. About 1920 lacquer started being used in production finish rooms. By 1940, shellac had almost completely been replaced by it.
So prior to 1920 - probably shellac
1920 - 1940 - could be either
1940 - present probably lacquer

There are some modern finishes such as water-borne, conversion and UV finishes, but they are relatively new and relatively rare in non-contract furniture.

As finishes degrade, they become less moisture resistant, so I would not rule out the finish just being old lacquer, or it could be shellac that displays similar propensities as it ages.

Before you embark on any of the following, try out your products and procedures on an obscure spot such as the underside of the table or pedestal in a hidden spot.

Quick Fix: You may buy some time by rubbing out the top with a cloth made for removing moisture. The same product is also available in a yellow and black package but branded Jasco. One or two cloths and about an hour's time will do your whole table and a couple of leaves.

A more durable approach, assuming you've ruled out Pledge, is to do a thorough cleaning. First use Dawn and water to get off any residual water-soluble soiling. Then wipe with mineral spirits to get off any non-polar residues (waxes, grease, mineral (AKA lemon) oil furniture polishes, etc.) Let dry. If you have some blushing because of the water cleaning, you'll have to back up to Quick Fix to remove the blush. These cloths will also ameliorate the silicone damage to some degree. If you use the cloths, re-clean afterwards with mineral spirits as they contain a rubbing oil.

If desired, at this point, you can do a light sanding with P400 sandpaper or Scotch-Brite pads. Just enough to dull the finish Don't use steel wool as it can leave metal shards that will dwell in your new finish.

You can then brush on some brushing lacquer such as Deft Clear Wood Finish or Watco Lacquer. Be sure it's a brushing lacquer as most lacquers are meant to be sprayed and will dry way too fast to brush on.

You may want to apply a second coat of lacquer after the first has dried.

You can spend some time trying to determine the finish that's on there, but the lacquer will bond with either shellac or lacquer assuming it's clean and dry.

I'm curious that no one has suggested polyurethane, possibly over Sealcote. Because the table is being used for meals. I'd expect poly to better resist abrasion and moisture.

Thanks so much for the advice. Klem1 pegged it that I am an amateur and I wouldn't attempt removing the existing finish myself. I'm also reluctant to take it to someone (unless one of you happens to be in Denver???). I think I'm leaning towards bobsmyuncle's suggestion of lacquer. I will do some experimenting with cleaning first and then work myself up tackling the lacquer. I also appreciate the suggestion of the cloths, as they seem like a good thing to have around in general, and the description says they can also be used on brass and I have a couple of brass items that could use some cleaning.

Btw, does anyone have experience with Restor-a-finish? The dining chairs have some wear and scratches and I was thinking of trying that product to give them a quick makeover.