As you correctly note, the solution is slightly different when the applied forces are not equal. Instead of deforming at rest, the rod deforms while accelerating. These concepts are illustrated by superposition.
The rod accelerates according to Newtons Second Law in the direction of the larger force. When forces are unequal, the rod is not in static equilibrium- the net force acting on the rod is the difference in applied forces. $$\therefore F_{more}-F_{less} = ma$$
The rod deforms from the applied forces, where axial deformation results from the internal force. $$\delta = \frac {F_{less}L}{EA}$$