We study spatial correlations and structure factors in a three-state stochastic lattice gas, consisting of holes and two oppositely “charged” species of particles, subject to an “electric” field at zero total charge. The dynamics consists of two nearest-neighbor exchange processes, occurring on different times scales, namely, particle-hole and particle-particle exchanges. Using both Langevin equations and Monte Carlo simulations, we study the steady-state structure factors and correlation functions in the disordered phase, where density profiles are homogeneous. In contrast to equilibrium systems, the average structure factors here show a discontinuity singularity at the origin. The associated spatial correlation functions exhibit intricate crossovers between exponential decays and power laws of different kinds. The full probability distributions of the structure factors are universal asymmetric exponential distributions.