Please clarify the exact difference between:
I've seen your hands before
I've already seen your hands
Which sounds more natural in everyday English and which sound more common in formal talking or writing?
It's not possible to say which is natural in everyday English as both sentences have different meanings in different contexts.
In case of human hands;
I've seen your hands before means the person has seen her or his hands before. This could mean the person remembers the hands or at least remember to have seen them at least once.
On other hand, I've already seen your hands means the person has already seen her or his hands.
My personal opinion is that you could use the latter one when you have seen the hands recently. Otherwise, use the former sentence.