Post-starburst galaxies (PSBs) have recently and rapidly quenched their star-formation, thus they are an important way to understand how galaxies transition from star-forming late-types to quiescent early-types. The recent discovery of large cold gas reservoirs in PSBs calls into question the theory that galaxies must lose their gas to become quiescent. Optical Integral Field Spectroscopy (IFS) surveys have revealed two classes of PSBs: central PSBs with central quenching regions and ring PSBs with quenching in their outskirts. We analyze a sample of 13 nearby (z < 0.1) PSBs with spatially resolved optical IFS data from the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey and matched resolution Atacama Large (sub-)Millimeter Array (ALMA) observations of $^12$CO(1-0). Disturbed stellar kinematics in 7/13 of our PSBs and centrally concentrated molecular gas is consistent with a recent merger for most of our sample. In galaxies without merger evidence, alternate processes may funnel gas inwards and suppress star-formation, which may include outflows, stellar bars, and minor mergers or interactions. The star-formation efficiencies of the post- starburst regions in nearly half our galaxies are suppressed while the gas fractions are consistent with star-forming galaxies. AGN feedback may drive this stabilization, and we observe AGN-consistent emission in the centers of 5/13 galaxies. Finally, our central and ring PSBs have similar properties except the ionized and molecular gas in central PSBs is more disturbed. Overall, the molecular gas in our PSBs tends to be compact and highly disturbed, resulting in concentrated gas reservoirs unable to form stars efficiently.