I finally picked up a cheap piece of pre-fab trim from the local craft store. It did not take the stain as well as the unfinished wood drawer units, but it'll do.
I'm not totally pleased with the final effect. The trim in the middle is too light and the stain did not take evenly on the drawers. The bottom drawer is particularly awful. So I'd suggest painting stuff like this instead.
Speaking of painting, I had fun obsessing over faux Delft drawer pulls: I painted the already attached knobs with white acrylic paint, painted a blue blob, and then put a couple coats of clear acrylic gloss.
They are pretty primitive, but because the design is simplistic, I could reproduce it three more times.
I used cardstock to line the drawers. It is easier to handle than paper is and glues in place with out warping or anything.
With the doll, for scale.