Remove the hardware and drawers. This will allow you to do a better job cleaning the edges of the drawers, as well as the edges around the drawer openings.
Clean the chest and drawer fronts with a household cleaner and a damp dishcloth or use 2 tablespoons of gum turpentine and 4 tablespoons of boiled linseed oil in a gallon of very hot water. Be careful not to flood the wood with water. Too much water could soak into exposed areas of the wood and cause it to swell, cloud the finish (if you allow it to sit) and cause the joint to loosen because the water will also work on the old glue.
Get rid of light scratches, stains and gummy buildup caused by years of oils and furniture polishes gathering dirt. Mix 1/4 cup gum turpentine and 3/4 cup boiled linseed oil in a jar with a lid and shake it up, then pour a few tablespoons into a jar of hot water. Dip the steel wool into it, then rub the steel wool back and forth on the wood, working with the grain.
Dry and polish the dresser with a soft clean cloth after cleaning.
Rub the entire clean dresser with a mix of 3 tablespoons of boiled linseed oil and 2 tablespoons of turpentine with a quart of hot water.
Clean the hardware with toothpaste and a soft cloth, then rinse and dry.
Remount the hardware and reassemble the drawers.