Yes, a full moon was important, as well as acceptable weather conditions. However, the state of the tides was also important. The planners wanted the landing to take place before dawn, on a rising tide. Not only did that expose more of the beach obstacles, it meant that grounded landing craft would be afloat again sooner, making them able to bring in reinforcements sooner. And, if the landings didn't take place on June 6, the next time the tides would be right would be in two weeks, without a full moon.