Studies on movement vs. lunar phase show small effects: slightly more midday daylight movement during full moons, slightly more dawn-dusk movement during new moons, and no measurable shift in actual rut breeding dates. The "rutting moon" theory hasn't held up against breeding-date data.
Don't reschedule a rut hunt around the moon. Schedule it around the photoperiod-locked peak window in your state and the cold-front forecast within that window. See state rut dates for the calendar.