We utilize James Webb Space Telescope (JWST) Mid Infrared Instrument (MIRI) integral field unit observations to investigate the behavior and excitation of H2 in the nearby Seyfert galaxies NGC 3081 and NGC 5506, both part of the Galactic Activity, Torus, and Outflow Survey (or GATOS). We compare population levels of the S(1) to S(8) rotational H2 emission lines visible to JWST/MIRI spectroscopy to models assuming local thermodynamic equilibrium (LTE), in order to estimate the column density and thermal scaling of the molecular gas. For the nuclear regions, we incorporate Very Large Telescope Spectrograph for INtegral Field Observations in the Near Infrared (or VLT/SINFONI) K-band observations to estimate population levels for available rovibrational H2 emission lines, and compare the resultant population curves to non-LTE radiative transfer models and shock modeling. We report a differing set of prominent active galactic nuclei (AGN)-driven excitation mechanisms between the two galaxies. For NGC 3081, we find that a non-LTE radiative transfer environment is adequate to explain observations of the nuclear region, indicating that the primary mode in which the AGN transfers excitation energy is likely irradiation. We estimate the extent of AGN photoionization along the ionization bicone to be ≍330 pc. In contrast, for NGC 5506, we find a shock scenario to be a more plausible excitation mechanism, a conclusion bolstered by an observed spatial correlation between higher-energy rotational H2 and [Fe II]5.34μm emission. In addition, we identify potential nuclear H2 outflows resulting from an interaction between the ionization bicone and the rotational disk. By isolating the outflowing component of the H2 emission, we estimate the warm molecular mass outflow rate to be 0.07 M⊙ yr−1.