Katie
When I clean my hardwood floors, or laminate floors all I ever used was a damp cloth with vinegar and water. Give that a try with your Swiffer mop when you have an empty container, simply put about 4 ounces of vinegar to 4 cups of water. This should let your floor dry on it's own without the water marks. (1ounce vinegar per cup of water)
Please keep me posted. TreeFrog