If you plan to resell the antique furniture, anything you do to the finish may or may not affect it's value. If the furniture is just old and nice but not actually an antique or terribilly valuable, then that is another story.

I'm wondering what's changed since you first got these pieces. Is it more humid than it's ever been before? Did you move to a new place with them? Have you gotten more sensitive? I wonder, too, if it's old mold. If it's the latter, that smell will likely never go away. I have a beautiful antique Mexican hutch, and it has The Smell. It's a choice. Keep my beloved piece and get that warm, woody smell as I walk by, or get rid of it. I wouldn't have that piece in my bedroom, for instance, but in it's current windy location, it's fine. So far, I haven't parted with it. May have to in the future, but for right now, this is my choice.

I once bought an antique dresser at an out of town outdoor sale and did not realize that it had an odor until I got it into my house. I washed it and I used tea tree oil to kill any germs. It didn't help. My husband sealed ALL unsealed surfaces (inside, underneath, drawers) with a water based poly finish and that solved the problem. I could not resell the dresser as an antique, but I'm OK with that.