"I did not use a tripod but I used a 300 mm"

There's your answer to the image being not sharp. Regardless of shutter speed, it's tough to hand-hold a 300mm lens and expect tack sharp images.

The Moon is in full sunlight, so there's plenty of light on the subject. There is a common modification to the "Sunny 16" rule for shooting the full Moon - f/11 at 1/ISO. 1/500 at f/5.6 is close enough, maybe a little underexposed.

Most lenses have extra room past the "infinity" setting to allow for expansion/contraction of the system at the temperature extremes.

Your autofocus system might have problems because the full Moon is inherently low contrast - there are no shadows from the craters and other surface features because it is in direct overhead lighting.