We demonstrate the importance of considering correlations between degenerate Zeeman sublevels that develop in dense atomic ensembles. In order to do this, we develop a set of equations capable of simulating large numbers of atoms while still incorporating correlations between degenerate Zeeman sublevels. This set of equations is exact in the single-photon limit and may be interpreted as a generalization of the frequently used coupled-harmonic-oscillator equations. Using these equations, we demonstrate that in sufficiently dense systems, correlations between Zeeman sublevels can cause nontrivial differences in the photon-scattering line shape in arrays and clouds of atoms.