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abstract: 'In the previous papers in this series, we have measured the stellar and [ <span style="font-variant:small-caps;">HI</span> ]{} 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 provide a measure of the vertical dark matter potential. We fit dark matter halo models to these potentials. Using our [ <span style="font-variant:small-caps;">HI</span> ]{}self-absorption results, we find that a typical dark matter halo has a less dense core ($0.094\pm0.230$M$_\odot$/pc$^3$) compared to an optically thin [ <span style="font-variant:small-caps;">HI</span> ]{}model ($0.150\pm0.124$M$_\odot$/pc$^3$). The [ <span style="font-variant:small-caps;">HI</span> ]{}self-absorption dark matter halo has a longer scale length $R_c$ of $1.42\pm 3.48$kpc, versus $1.10\pm 1.81$kpc for the optically thin [ <span style="font-variant:small-caps;">HI</span> ]{}model. The median halo shape is spherical, at $q=1.0\pm0.6$ (self-absorbing [ <span style="font-variant:small-caps;">HI</span> ]{}), while it is prolate at $q=1.5\pm0.6$ for the optically thin. Our best results were obtained for ESO274-G001 and UGC7321, for which we were able to measure the velocity dispersion in Paper III. These two galaxies have drastically different halo shapes, with one oblate 