Mineral spirits only softens the wax, it does nothing to remove it.

You have to use rags (and a lot of them) to soak up the softened wax and mineral spirits.

There are solvents that can actually break down the wax, but they will destroy just about any finish present.

Buy a box of non-woven rags and apply mineral spirits with one rag, let it set a minute or two, then wipe it off with another rag.
Turn and fold the wiping rag after every swipe to have a clean surface.

Thanks so much Bobsmyuncle and Brickeyee! I really appreciate the advice and timely responses.

I had seen Jeff Jewett's article. Like you said, it is very good at explaining. Since he starts with mineral spriits, I thought I would too. I'll definitely try the dawn and water approach. It's also good to know that letting the solvent sit for a minute (increasing the time factor) won't hurt the underlying finish. I'll try that too.

Regarding temperature - we keep our house a bit on the cool side - about 65 degrees. Would it help if I turned up the temperature while I did this? Or would a 5-10 degree increase not make much difference?

I'll let you know how it goes as I continue. Thanks again very much!!

Be careful using mineral spirits indoors in the winter.

They are flammable, as is the vapor.

It is also narcotic if you inhale enough of them.

All turning up the temperature will do is evaporate them faster.

I would actually use them outside (even when cold) since it would slow the evaporation and the wax would stay dissolved longer.

Keep the can warm in a cooler.

Good point. We don't want to gas ourselves out. Luckily, we live in SE Virginia and, if this cold ever breaks, do get days where we can open the windows even in the winter. Depending on how it goes, I might not finish until spring, but at least our dining room is one where we can shut all the doors to keep the fumes out of the rest of the house, and set up a big fan in the window. Keeping the can warm in a cooler is a good idea!

Thanks!!!

A household cleanser like ammonia takes care of wax quickly, but it's very hard on finishes as well. I use Fantastic cleanser and steel wool for de-waxing, or naphtha if outdoors. For some reason I like naphtha better than paint thinner solvents.
If you actually must get every speck of wax from an area, I suppose you must use something that breaks it down, like ammonia; but if you were going to re-wax the article in question, just getting most of the buildup off with solvent, rags and a good buffing with a woolen cloth would be perfectly acceptable.
I think your paneling would respond very well to being waxed with a colored wax like Briwax.
I have to stop typing "wax" now; geez!
Casey

A household cleanser like ammonia takes care of wax quickly, but it's very hard on finishes as well. I use Fantastic cleanser and steel wool for de-waxing, or naphtha if outdoors. For some reason I like naphtha better than paint thinner solvents.
If you actually must get every speck of wax from an area, I suppose you must use something that breaks it down, like ammonia; but if you were going to re-wax the article in question, just getting most of the buildup off with solvent, rags and a good buffing with a woolen cloth would be perfectly acceptable.
I think your paneling would respond very well to being waxed with a colored wax like Briwax.
I have to stop typing "wax" now; geez!
Casey

To be honest, I'm not quite sure what I'm going to do when I am done. There is some damage to the finish in the way of small scratches and rings. I figured I'd clean it really well and then see how it looked before I decide. I'm not sure what the finish is. Depending on what it is, I thought of maybe trying to reamalgate it.

If the finish needs to be redone, then I have to think about what to use that will resist scratching but still look good with the sides that I am not working on. But that's a subject for a different thread. I think I'll start that one once I know what I'm looking at.

Ammonia will remove some finishes....not just damage it but remove it.
Stick with the mineral spirits and elbow grease....it's safer.
I suspect the clouding is wax, but it could also be a lacquer finish that has deteriorated and the mineral spirits is softening it and removing it.
To see if you have a lacquer finish, get some lacquer thinner and test a small spot in a hidden spot with a Q-tip.

Linda c

Interesting progress - I tried letting the mineral spirits sit for about 10 minutes in one area. I thought I was pretty smart about it too - I poured on a little and covered it with plastic wrap to keep it from evaporating, and it helped make it a nice big thin film over the surface. (I'm pretty proud of myself for that one!) It covered an area maybe 8 x 8 inches.

After letting it sit, I started at it with the 0000 steel wool, adding more mineral spirits and wiping frequently with a cloth. I was amazed at how much black gunk came off! And how different the wood looks under all of it! It is redder now, and much more of the detail in the grain is visible. I'm pretty sure I didn't accidentally hit bare wood because a nice shine is still there when it dries. The good news is that I don't have to scrub nearly as hard. It still takes some elbow grease, but cleaning the whole top at least seems possible now! And I feel better about not scrubbing the finish raw.

I really am amazed at the difference. I never thought it was that dirty. The surface feels the same - the dirty areas are also nice and smooth, no gumminess/greasiness etc.

Now I just need to wait for it to warm up a bit on a weekend so I can open the windows and do the rest! Thanks so much everyone for the advice!