Remove all the items in the dresser drawer. Take your soap and water mixture and slightly damp your cloth. Clean out any dust and debris in the drawer. Allow the drawer to dry.
Pull the drawer straight out of its holding. Stop pulling if the drawer is stuck. Push the drawer inward, and then gently pull back out. This should remove the drawer from its holding.
Check to see if your dresser drawer has a latch on the side of the drawer. If so, push the latch to release the drawer, or if the latch is screwed in, unscrew it, and pull the drawer out fully.