Flip the drawer over too, as a true period piece will not only have a solid wood bottom, but that wood will be thick and beveled to fit into the grooves of the frame of the drawer. High gloss finishes and polyurethane are 20th-century elements.
Use your fingers on drawer bottoms or backboards of case furniture.
Note whether the drawer has dovetails, and whether they are machine-cut dovetails or hand-cut. This is usually a sign that the piece is indeed antique—and high-quality at that, since rear dovetails are very uncommon even in antiques.
Drawers (and backs) are also usually one of the cheapest components in furniture. Most modern pieces—even high-quality reproductions that look very genuine on the outside—use plywood in drawer construction.
Nuts are more common for antiques, while screws are a newer convention.
Also check to see if the hardware has been replaced: usually there will be marks or holes on the wood around the hardware.