You might take a wait and see, until the wood has for certain reached it's original, pre-disaster moisture content.
There are lots of ways to tackle this. The white discoloration can be tackled with a "water ring remover" product.
The areas where the stain seems to have been affected really need to dry out completely. It may be that these areas would respond to a tinted wax or could need to be refinished and carefully color matched; but advise like that can't really be dispensed online, it needs to be seen in person.
Casey

What's a tinted wax? Those wax sticks you can buy in various colors and look like crayons? Thanks!

Thanks Casey.

Let say I need to call a professional what kind of tradesman you think would be best suited for the job?
I have been calling carpenters and painters and nobody looks interested.

We found a person who specialized in commercial-sized wood finishing projects once. He was like a specialty painter. You might check with furniture stores and see who does their touch-up work, or with a well-established antique dealer or auction house.
As per your other post elsewhere about polyurethaning it; My advice is don't use anyone who suggests poly. It was not used originally, and will not be the best compatible material for any repairs.
Casey

thanks for the advise, Casey. I am calling people around and I have no luck. I am not very knowledgeable about the subject. I though you generally stain and then poly. I hope I can get some professional with experience.

Polyurethane is one type of varnish. Another type of varnish would have been used originally; either shellac (so-called "spirit varnish") or a non-poly oil varnish. It's a remote possibility that the paneling was lacquered originally. A good finish man can tell you what you have and the best way to repair it. Stripping is a last resort.
Casey

You might be able to remove the finish blush by padding with a cloth dampened in denatured alcohol. If this is the original finish from 1900, there is a good chance it is shellac. The alcohol will dissolve the top layer and allow the moisture to escape. Just go lightly as it will remove the shellac if you get too heavy on rubbing or too wet. Test in an obscure spot before starting out on the middle of the panel.

Here are some

and pictures of a barrister's bookcase that sat in a leaky moving van under a mover's blanket for a couple of weeks. A quick wipe with a rag made the blush disappear immediately. I followed up with another wipe of a pad dampened in shellac.

bobsmyuncle: that's impressive. I ll give it a try. I have so much of this paneling in my home that there in no problem finding a hidden part.

I kept calling all day today but I had no luck. I got to talk to a decorated "restoration expert" (highly advertised) but he was only interested in striping and starting a new. No interest at all to do a "patch" job.

Update:

I tried denatured alcohol and it worked really well for all areas that were foggy. It was easy so I feel comfortable that I can finish the job myself.

Now I am left only with the areas that have the deeper discolored lines.