Old varnish may have absorbed odors, and old shellac often emits a sharp odor when wet (or when scraped and sanded off).
I never equated that with a BO. Solution would be to keep it dry. Does the smell go away after drying? You could wax the woodwork, which will impart its own smell for a while, maybe mask the odor. A new coat of finish will seal in any odors.
Casey

Well, yes, it goes away after drying, but it needs to be dampened when I clean it. At least for this initial, intense cleaning. I look forward to the day when I can just dust my woodwork and call it a day. Right now, there's decades of filth that needs to be removed!
The finish is alligatored, or so says the interior decorator at the local paint store. It looks like the finish beaded up on the wood decades ago. I googled "alligatoring" and that's not what my wood looks like. Anyway, I'm hesitant to add another layer of anything, given the fact that I may someday tackle the job of restoring the wood work. I'd be creating more work for myself if I added another layer, right?
Friends have offered many other colorful ways to describe the scent. It's probably the worst thing I've ever smelled in my life. Sigh.

Could be smoke from previous owners.

Any chance it's stale urine from past pets?

Yuck. There was dog hair stuck to things, so maybe it's urine. I'm not sure if the wood above my head smells. I've been cleaning what I can reach for now. I suppose if the higher wood doesn't smell, the dog will become the culprit. I'll check out enzyme cleaners next time I'm out.
Thanks!

LOL Katie. It's not that big a thing, in that house pets are common, and even the best of them often find a spot where they go when out of sorts, and often the owners don't even know it, until a carpet comes up. Woodwork is a fav venue for male dogs, since they prefer to hike a leg up. Sometimes......no make that often, it will soak behind a baseboard, making it rather difficult to get to. Another spot you need to address, not only for pet pee, but God knows what else, are the register runs. I have found things you don't even want to know about in register runs......and always clean them out as far as I can reach with disinfectant. If you are allergy prone, it's not a bad idea to have your runs, the entire length of them professionally cleaned.

Alligatoring looks lots of different ways - scales, skin, grasscloth, even. can be really beautiful, actually.

with the smell, I've come across wood that had soaked up all sorts of odd cleaners and treatments - naptha, bleach, kerosene, vinegar...

waxing might be the best stop-gap measure to protect the wood on the one hand, and your sinuses on the other.

something to try might be to infuse some oil (I prefer olive oil with a little beeswax melted in to it, some folks swear by mineral oil) with an astringent herb (rosemary, lavander, thyme, even black pepper essential oils work well) and rub the wood down the day before you wax it.

: ) welcome to your new old house, by the way - we just finally got done the 'clean, paint, unpack' stage ourselves (had to gut the powder room, found an oak floor under the vinyl) so I can sympathize !

The worst possible reason for the bad smell is mold contamination, which can be very difficult to remove. The fact that it's worse when damp is consistent with mold contamination. If there was a plumbing leak inside a wall that was allowed to continue because it was not detected, that would produce a persistent mold problem.

If the smell is detectible in ALL the woodwork (I assume you mean baseboards) throughout the house, mold is not likely the cause, unless an animal urinated all over the place and nobody cleaned it. But if the smell is more pronounced near plumbing, that would raise the suspicion that the origin of the smell is mold.

I hope it is not mold, because mold remediation is complicated and expensive -- could easily cost $2 thousand dollars or more.

Haus, I'd be sick if it was mold! I'm highly allergic. Thanks for the info, though. Good to know.
Chinacat, congrats on your house. :) I like the idea of infused oil. I'll keep it in mind!