We theoretically investigate an electric-field-driven system of charged spheres as a primitive model of concentrated electrolytes under an applied electric field. First, we provide a unified formulation for the stochastic charge and density dynamics of the electric-field-driven primitive model using the stochastic density functional theory (DFT). The stochastic DFT integrates the four frameworks (the equilibrium and dynamic DFTs, the liquid state theory and the field-theoretic approach), which allows us to justify in a unified manner various modifications previously made for the Poisson-Nernst-Planck model. Next, we consider stationary density-density and charge-charge correlation functions of the primitive model with a static electric field. We predict an electric-field-induced synchronization between emergences of density and charge oscillations. We are mainly concerned with the emergence of stripe states formed by segregation bands transverse to the external field, thereby demonstrating the following: (i) the electric-field-induced crossover occurs prior to the conventional Kirkwood crossover without an applied electric field, and (ii) the ion concentration dependence of the decay lengths at the onset of oscillations bears a similarity to the underscreening behavior found by recent simulation and theoretical studies on equilibrium electrolytes. Also, the 2D inverse Fourier transform of the correlation function illustrates the existence of stripe states beyond the electric-field-induced Kirkwood crossover.