My brand new 5D Mark III - what am I doing wrong, please help!

[Enrico]

And another thing worth pointing out (that you praobably knew) is that the DOF - Depth of Field gets alot narrower on a FF like 5D3 than on a crop body.

When I moved to the 5D3 I was dissapointed with the wide open shots in the beginning not being sharp... but if I looked closely I could see where the focus was

IE. I had to start paying more attention on where I put the focus (than on the crop body) AND I had to AFMA all my lenses. 135/f2 not tuned perfect to your new body will give images out of focus on f2.

You might also have had the settings on your crop body JPEGs being boosted with brightness, contrast, sharpnes etc whereas you don't have that on the 5D3 as of now.

 

[Kernuak]

And another thing worth pointing out (that you praobably knew) is that the DOF - Depth of Field gets alot narrower on a FF like 5D3 than on a crop body.

Actually, that's only true if you have the same field of view, because then you are either using a longer focal length or are closer than if you were using a crop sensor camera. If you are standing in the same place with the same lens (which is likely in a gym/sports arena), then full frame actually has greater depth of field.