I would try the Howard Restor-a-finish.

They do make other products, which dissolve the old finish and make it appear brand new, probably not what you want.

Here is a link that might be useful: Howard Restore a finish

Never heard of it, so I looked:

Revitalizes damaged wood-stained surfaces and restores their original colors and lusters
Simple wipe-on, wipe-off application removes grease, grime, white heat rings, smoke damage, oxidation and watermarks
Removes scratches and blemishes

Here is the kicker:

Oil-based solution contains just enough solvent to penetrate wood finish without dissolving it

Know what that stuff is?

Hockey puckey.

Why? Because there are no common oil based finishes that can be dissolved by another oil based product.

In fact, the only common finishes that can be dissolved by a solvent or more finish are lacquers and shellacs.

That crazing/etc. will have to be removed to fix, unless it is lacquer or shellac. And even then it looks to be too old and thick for lacquer thinner(for lacquer) or denatured alcohol(shellac) to affect enough to fix with out removing all of the stuff.

Based on the drawer construction, from what I have read, this is from 1870~1900.

Not sure exactly what was used back then, but I believe it to be shellac.

That is a good guess. The way to find out is to wipe alcohol on the finish. If any comes off, it is shellac. However, I have read(have not tried removing any finish that old) the really old shellac is very difficult to remove with just alcohol.

Plus, there is a good chance another finish has been applied. Crazing does happen when dissimilar finishes are applied to each other.

There are ways to re-amalgamate crazed shellac finishes. It looks to me as if a new coat of shellac was applied 50 years ago over some dirt and oil or wax, so the re-coat was not as firmly anchored to the original and it crazed.
The process is not foolproof, and this is not the piece to learn on.
It involves a thorough cleaning, a fine wet-sanding down of the alligatoring, and then re-polishing with shellac. If done properly, you still have the original color, original finish, and a new topcoat of shellac, which can be rubbed out of waxed, etc.

But it takes a few pieces worth of practice to get the tricks all down pat.

This piece isn't dark or discolored, and all the finish appears to be intact, so my advice is to leave it be.
Casey

Old shellac sometimes gets a crazed appearance all by itself.

If the finish is shellac you can try brushing lightly with some denatured alcohol and see it it will dissolve, smooth out, and then harden.
The hardening can be an issue for some very old shellac.
It will dissolve and smooth out, but like new pre-mixed shellac that is old, can no longer form a hard surface.
It just stays 'sticky' and soft.

Test in an inconspicuous area first.

You should probably go over the surface with paint thinner and a lot of soft paper towels to clean it as much as possible.

Paint thinner will not harm shellac, and can remove oils, waxes, and surface dirt.
This keeps the finish clean when you re-dissolve it.

Adding another coat of an appropriate grade of shellac (if the old stuff smooths and hardens) without bothering to remove the finish will work fine.
The new layer will 'melt' into the old, as long as the old can re-harden.

I wiped some denatured alcohol on a spot. Not really anything on the crazed surface.
I wiped it on the non-crazed surface and it got tacky and I rubbed and smoothed out.

Maybe this is a second finish on this?
Or maybe the shellac is much thicker on the crazed surface.
Anything that will soften shellac better then denatured alcohol?

Shellac is made from alcohol, so that is the best remover.

Since you have two different finishes, you may want to use a furniture refinisher solution (glorified nail polish remover), to remove both.

"Anything that will soften shellac better then denatured alcohol?"

Not without destroying the shellac.

Shellac is a resin secreted by the female lac bug, on trees in the forests of India and Thailand. It is processed and sold as dry flakes (pictured at right), which are dissolved in ethyl alcohol to make liquid shellac, which is used as a brush-on colorant, food glaze and wood finish.

http://en.wikipedia.org/wiki/Shellac

Alcohol dissolves the dry shellac flakes. Once the applied shellac has cured(all the alcohol evaporated, another application will partially dissolve the first layer. Age makes little difference until many years have passed.

You could make shellac whisky by dissolving shellac flakes in grain alcohol. I have actually tasted the dry flakes I buy to mix my own finishes and preconditioners. Really no taste for me.

If that is oil varnish over the original polish, then there is no saving it, you'd have to strip it.
Casey

Try mixing shellac thinner (denatured alcohol) with 1/3 part lacquer thinner...
Drip or slosh a little on an inconspicuous spot....
I will bet it will dissolve the chequered finish.....and perhaps allow it to smooth out.
You may have to use the lacquer and shellac thinner to dissolve the finish and wipe a lot of it off....leaving just a sealer coat....allow to dry several days and re-coat with shellac.
Linda C