What has already been said, but I would add to use a proper plug and not a dowel as the filler. a proper set of plug cutters allow you to cut wood plugs of the same type wood and grain, and with a good wood glue it becomes an invisible, and clean job. The plug cutters can allow you to go oversize, which facilitates good purchase by the screw, and no interference with the screw threads, and the plug joint.
Bill