Zack Fair's last ability doesn't cause you to move counters from Zack Fair onto the target creature. Rather, you put the same number of each kind of counter Zack Fair had when it was sacrificed onto the target creature.

In some unusual cases, you may end up putting the appropriate counters on more than one permanent. For example, if you control The Ozolith ("Whenever a creature you control leaves the battlefield, if it had counters on it, put those counters on The Ozolith.") when Zack Fair's last ability resolves, you'll put the appropriate number of each kind of counter onto both The Ozolith and the target creature.

Zack Fair's last ability puts all counters that were on Zack Fair onto the target creature, not just its +1/+1 counters.