I'd guess circa 1940's. Restoring is a problem and perhaps not worth the expense. I would place the 'newer' pulls on the bottom drawer. (Less Used).
As far as the molding, Home Depot, Lowes, etc. sell molding which may or may not match, or perhaps come close. Replace with that which come close.
Rub down with mineral spirits. Rags and elbow grease. Lots of elbow grease. Apply a stain that is complimentary to the piece or to your liking. Minwax sells a variety of stain/finishes.
You should be able to do all this yourself for less than $50. You'll end up with a decent dresser.