[sbuczkowski]

While it sounds like most of your sharpness issue may be related to hand-holding the 300mm lens, don't forget that the moon is a moving object. Motion is magnified along with the image when using a long lens.

For a full sense of this motion: the moon subtends half a degree of arc in the sky and 'moves' at about 15 degrees per hour. The full moon will move it's own width every two minutes.