Don't pack dresser drawers, movers usually move dressers with contents inside. Label and securely tape all cartons. Try to avoid open cartons. Do not pack flammables, combustibles or explosives, spray cans, gas, etc. Movers cannot take these hazardous items.
Many people save and use their old newspapers.
Keep in mind, however, that the ink on newsprint never thoroughly dries. Consequently, the goods you wrap in newspapers are most likely going to be soiled and will require cleaning after unpacking, and before you put them away. For items you prefer to keep clean it would be best to purchase some packing paper from your mover. You'll need many boxes in assorted sizes. All boxes should be in good condition, and must have covers on them in order that they can be closed up and sealed.