Why do atoms bond? Why are unfilled valence shells energetically unstable?
The idea of a "full" outer shell is a bit of a misnomer, for instance 3rd row has 3d orbitals but doesn't bother to occupy them until 4th row. Does Argon have a full outer shell, even though it has a $3d^0$ configuration?

Generally there's a large energy gap between the p-orbitals in one row and the s-orbitals in the next row, because of the increase in shielding and such factors, which acts as a kind of natural barrier, or a step, to adding any more electrons than what appears to be a 'full' valance shell.

But within any row, the amount of shielding and associated factors (penetration, nuclear charge, etc) remain largely similar, so if the valence shell isn't 'full' there is still sufficient attraction between the electron and a neutral atom that a negative ion may be stable.  Adding more than this, the energy gap is too large to overcome under usual circumstances so it stops at a 'full' shell.

There are always exceptions.  Noble gases lower down can form compounds even though they are "stable" "full" outer shells, and quirks like some transition metals having $s^1d^{10}$ configuration go to show that some of the ideas about electron filling and stable shells are rules of thumb, each individual case really needs to have its energy levels calculated to explain them.