In previous papers in this series, we measured the stellar and H I content in a sample of edge-on galaxies. In the present paper, we perform a simultaneous rotation curve and vertical force field gradient decomposition for five of these edge-on galaxies. The rotation curve decomposition provides a measure of the radial dark matter potential, while the vertical force field gradient provides a measure of the vertical dark matter potential. We fit dark matter halo models to these potentials. Using our H I self-absorption results, we find that a typical dark matter halo has a less dense core (0.094 ± 0.230 M⊙ pc-3) than that for an optically thin H I model (0.150 ± 0.124 M⊙ pc-3). The H I self-absorption dark matter halo has a longer scale-length Rc of 1.42 ± 3.48 kpc, versus 1.10 ± 1.81 kpc for the optically thin H I model. The median halo shape is spherical at q = 1.0 ± 0.6 for self-absorbing H I, while it is prolate at q = 1.5 ± 0.6 for the optically thin case. Our best results were obtained for ESO 274-G001 and UGC 7321, for which we were able to measure the velocity dispersion in Paper III. These two galaxies have very different halo shapes, with one oblate and one strongly prolate. Overall, we find that the many assumptions required make this type of analysis susceptible to errors.