  In the previous papers in this series, we have measured the stellar and \hi
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 \hi 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 \hi model
($0.150\pm0.124$\,M$_\odot$/pc$^3$). The HI 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 HI model. The median halo shape is spherical,
at $q=1.0\pm0.6$ (self-absorbing \hi), while it is prolate at $q=1.5\pm0.6$ for
the optically thin. 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 drastically 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.
