I don't understand how everyone is getting such slow shutter speeds. I usually get speeds between 1/60s and 1/125s at f/11>iso320>840mm. Shooting at 300mm and f/5.6 , you should have much higher speeds than me.
I usually get my best results with a shutter around 1/125s .......................This is more than enough to compensate for the movement of the moon.(it moves faster than you think, so shutters below 1/60s will show some blur, unless you get lucky)