Why doesn’t voltage drop to 0 when a capacitor becomes fully charged?
You have some misconceptions about the physics behind that phenomenon.

First of all, we are not talking about valence electrons here. The plates of the capacitor are metallic, hence most of the electrons of the outer shells of the atoms are in conduction band, not in the valence band. 

Atoms of metallic materials form a so-called metallic bond, where all the outer shells electrons form a huge orbital extending all over the metallic crystal lattice. In this sort of "big cloud of electrons" electrons are relatively free to move even without external forces applied, i.e. they are not bound to a specific atom.

When you apply a battery to the plates, it works like a sort of pump, that is it "moves" electrons from one plate to the other. The process doesn't stop when "the atoms cannot accept any more electrons", but when the difference of potential between the plates equates the voltage of the battery, reaching an equilibrium.

In fact, each electrons that is displaced from a plate to the other generates a little bit of electric field which is not neutralized by a corresponding positive charge in a nearby atom. As you already got it, it leaves the positive charge on the other plate. This means that the charge displacement creates an electric field in the space between the plates, and that field has a corresponding potential function, which in turn gives rise to that difference of potential between the plates.

The more charge you displace, the more the field is intense, and the more the potential difference is big. That potential difference opposes the voltage of the battery, and the difference of the two is the actual "voltage around the loop" which makes further electron move. As you have guessed, more potential difference across the plates means less voltage around the loop, hence less current (for a given circuit resistance).

In theory this process is infinite (I'll avoid the math here), i.e. that equilibrium is reached after an infinite time. Actually after 3-5 time constants you may consider the capacitor charged.