Clean it up with mineral spirits on a cloth...on lots of cloths because you will toss them when theya re dirty...paper towel is good too.
Unless it has been refinished, it's unlikely the finish is poly.
To get a new stain to "take" you will have to completely remove any old finish. I recommend a commercial stripper from your paint store.
You strip the pieces, sand everything, apply a new stain of your choice and apply 3 coats of finish, poly is good, rubbing with steel wool between coats.
OR...you could do quick and easy...and clean the pieces up and apply a coat of darker colored varnish.
BUT...when you chip or scratch the finish, it will show lighter beneath.
The best way is to strip, sand, stain and finish...but you could get by for a few years with just a new coat of varnish.
Linda C

I know a lot of professional finishers. Many of the best ones are women.

I mostly agree with LindaC's comments except:
- Using steel wool between coats is likely to leave steel shards that will eventually rust
- I hate the "colored varnishes" such as Polyshades. They are very difficult to apply without streaking and turning opaque.
- Most of the dirt will come of most easily with a water cleaning. This is fine as long as the finish is sound.

Here is a link that might be useful: Cleaning a finish

I've finished miles of wood surface and never had and steel wool shards left in the finish...in my fingers, yes but not in the finish. Of course I knew the possibility existed and so was very careful to remove all the stuff, use a tack cloth and wipe carefully
And I too agree the colored varnishes may streak....I have tinted varnish with artist's oils...because that's what I had at the time. I found it worked well and repeated it the second time I wanted to darken a finish. It's a quick fix.
I wish you would hang out some on the antiques forum and promote your ideas on finish preservation. There's someone there who believes any old finish should be removed, always, no question, and he says I am an idiot for suggesting that a lot of old finishes can be cleaned up and "patched" and waxed.
Linda C

Did I hear my name mentioned ?

Let's be clear about one thing, lindac is NOT a professional finisher.

Linda why would you use steel wool between coats ?
A light "cut" with a worn piece of sandpaper does the job, no tack cloth needed.

You called yourself an idiot not me. I don't like you because you run from forum to forum answering peoples questions like you are a Britanica Encyclopedia . Again, stick to what you know,it's clearly not wood types and how to refinish a piece of furniture.
Every time I confront you , you run and hide.
The last time you tried to upset me by posting my personal information to the forum, you were outed by half the members for setting up a fake screen name that was actually your son that you were using.

Again, I can't sell what your claiming is the answer to my customers.You don't seem to understand that.I can't guarantee my work if I were to go the route you always suggest.
Underneath that ladies ( you know the one)Walnut buffet and china is beautiful Walnut, why on earth would you want to leave it basically black with no ability to see the grain ?

Try answering just one of my questions ?
Like I said and have seen in the past, when confronted you run and hide.

Um. I'm staying out of this one.

Um, OK, thanks for the replies. I'm sorry, didn't mean to get a fight started here... LOL. My DH & I are going to tackle this project in a couple of weeks, so if there are any more suggestions, I'd love to hear them.

Thanks again!