Do full valence shells always result in a formal charge of zero?
The simple answer is no. For instance, the halide ions bear a formal negative charge with full valence shells.
Formal charge on an atom is decided by the number of protons in the nucleus, versus the number of electrons around the atom (or shared by that atom!). If the numbers are equal then the atom is neutral. Excess or deficiency of electrons result in a formal charge.
A good place to start would be the ozone molecule,