We present the results of Monte Carlo simulations of transport of charge carriers of a single type in devices consisting of a disordered organic semiconductor sandwiched in between two electrodes. The simulations are based on hopping of carriers between sites with a Gaussian energetic distribution, which is either spatially uncorrelated or has a correlation based on interactions with randomly oriented dipoles. Coulomb interactions between the carriers are taken into account explicitly. For not too small injection barriers between the electrodes and the organic semiconductor, we find that the current obtained from the simulations can be described quite well by a one-dimensional drift-diffusion continuum model, which takes into account the long-range contributions of Coulomb interactions through the space-charge potential. For devices with low injection barriers, however, the simulations yield a considerably lower current than the continuum model. The reduction of the current for uncorrelated disorder is larger than for correlated disorder. By performing simulations in which the short-range contributions of the Coulomb interactions between the carriers are omitted, we demonstrate that the difference is caused by these short-range contributions. We can rationalize our results by analyzing the three-dimensional current distributions and the in-plane radial distribution function of the carriers resulting from the simulations for different injection barriers with and without taking into account these short-range contributions.