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There's no direct relationship between formal charges and full valence shells. Full valence shells do no always result in a formal charge of zero, they can result in formal charges of -1,+1 and others. The formal charge is calculated by subtracting the sum of the number of lone pair electrons and the number of bonded electrons from the number of valence electrons of the specific atom.
The formal charge is the difference between the number of valence electrons "belonging" to the bonded atom and those in the full valence shell. And, no the formal charge is not always 0, but that is when it has the lowest energy.
Formal charge is the relative charge of an atom relative to other atoms within the same molecule. Valence electrons just resemble outer electrons and may or may not be used up as bonds in the molecule; it is used for calculation of the formal charge.
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