You can select a minimum shutter speed in Av mode, 1/125 would be appropriate, you would only be going to iso 3200, which is nothing high by todays standards and the 6d looks good up to about iso 12800!
Single shot, try working with inanimate objects first, don't recompose, and see if it is in fact the lens. You cannot do micro adjustments while you are in servo mode, it will not be accurate.
You will have a problem focus recomposing in servo mode, because as you recompose it will begin to hunt for focus again.
I never have a problem with focus and recompose on my 6d. People that say it is no good are the ones who's technique needs work.