So what exactly happens to the potential inside the resistor ?
Unlike the ideal conductors, for which an electric field cannot exist inside, there is an electric field through the resistor body when there is a current through.
And, as you may know, the rate of change in electric potential is related to the value of the electric field. Thus, the potential must change from one end of the resistor to the other.
The electrons moving through the structure of the resistor collide with the structure giving up kinetic energy in the process. The electric field within accelerates the electrons so that, on average, just as many leave the resistor as enter the resistor (charge does not accumulate inside the resistor).
Where does the electric field come from? The electron charge density at one end of the resistor is greater than at the other end. Indeed, the electron charge density must be greater on one of the conductors than the other since there is a potential difference (voltage) across the conductors.
In summary, there is an electric field through the body of the resistor that acts to 'keep the electrons moving' in spite of collisions with the resistor structure. Associated with this electric field is a change in electric potential.