Can you be more specific about what you want to accomplish here? '...change the wood on the bottom of the vanity' is a little vague.

I'm not sure how to fix it exactly
but I don't like the way the dresser rests on the wood
trim.
Possibly adding a draw at the bottom
but also keeping the nice old style of the dresser.

I think normally there should only be about 4" underneath a vanity or it would touch the ground and have a toe kick?

I took more photos to show the angle better.

If it was mine, this is what I would do:
- Remove the front piece with the curves in it. Maybe even the feet stile fronts next to it.
- Remove the base board on the wall starting where it goes under the existing unit.
- Extend the face frame down and include a 3-4" toe-kick.
- Fill in the intermediate area with one or two side-to-side drawers. This will not only look best but give you a bit more storage.
- Stain to match, or if you are not able to match stains, strip and refinish the entire front.

You're going to have to play it by ear; it may be easiest to remove the unit to make the changes, or not.

It appears a real Rube Goldberg at the moment and that would make it appear more usual.