Wood – Built wood vanity drawer deformed after painting!

There are several stain-like finishes for wood:

1. Penetrating stain - can only be used on unfinished wood. Several coats can be used to even the tone and make it slightly darker. Once the wood has a finish coat, it will only sit on the surface, and usually looks bad. (Most are solvent based, but there are also water based stains)
2. Varnish/poly stain - this is stain mixed with a finish coat, such as varnish or polyurethane. It mostly sits on the surface and can be darkened somewhat by adding more coats.
3. Pigmented stains - these are mostly used outdoors on decks and outdoor furniture. They are more like thinned down paint than true stains. The penetrate slightly, but mostly sit on the surface. They are not especially well suited to indoor furniture.
4. Colored glazes - these are somewhere between paints and stains, with a thick body that can be brushed or wiped on, then partially removed to give a glaze, a partially transparent color and finish. Often used to give an antique look. Often they are rubbed off at edges to let contrasting under layers show through.

Your furniture looks like it may have been done with varnish/poly stain. If this is so, more stain will not penetrate. If you sand off the excess, there will still be some areas that are effectively sealed because of the finish part of the coating. You might be able to add more of the same and get a more even coating. However, you may need to sand most of it off and effectively start again.

Surface stains are more prone to have lighter edges because they do not penetrate, and as you brush them out, very little stain and color is left on the edge.

When you say see a little bit of the grain do you mean texture or color variations. If texture, you are talking about a thinned paint-like finish, a type of glaze, rather than a true stain. Your cabinets are made of wood that has very little surface texture. if you mean slight color variations, either penetrating stain or varnish/poly stain can do this if carefully applied.

I've used the steel wool + vinegar method before*. It's essentially a stain so really only affects the surface layer. It shouldn't have any meaningful affect on the strength of the piece of wood. Splitting and warping is related to moisture and drying so wouldn't really be affecting by weathering stain.

After you weather it, note that most any additional finishes will change the color. With pine (and actual drift wood) I've found both water and oil based polys will give it a slightly darker, slightly yellow tint (varnish will typically add color as well). So definitely use test pieces before committing.

Rule-of-thumb is that water-based finishes will add less color than oil-based finishes. That said, if the goal is to 'age' the wood, perhaps no finish at all is suitable. Let it age even more in use. :)

PS (*) one trick is to pre-'stain' the wood with black tea. This puts more organic material into the surface of the wood for the oil and vinegar mix to react to.