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Previous research centered on the hydrodynamics in X-ray pulsar accretion columns has largely focused on the single-fluid model, in which the super-Eddington luminosity inside the column decelerates the flow to rest at the stellar surface. This type of model has been relatively successful in describing the overall properties of the accretion flows, but it does not account for the possible dynamical effect of the gas pressure. On the other hand, the most successful radiative transport models for pulsars generally do not include a rigorous treatment of the dynamical structure of the column, instead assuming an ad hoc velocity profile. In this paper, we explore the structure of X-ray pulsar accretion columns using a new, self-consistent, "two-fluid" model, which incorporates the dynamical effect of the gas and radiation pressure, the dipole variation of the magnetic field, the thermodynamic effect of all of the relevant coupling and cooling processes, and a rigorous set of physical boundary conditions. The model has six free parameters, which we vary in order to approximately fit the phase-averaged spectra in Her X-1, Cen X-3, and LMC X-4. In this paper, we focus on the dynamical results, which shed new light on the surface magnetic field strength, the inclination of the magnetic field axis relative to the rotation axis, the relative importance of gas and radiation pressure, and the radial variation of the ion, electron, and inverse-Compton temperatures. The results obtained for the X-ray spectra are presented in a separate paper.