Abstract Post-starburst (PSB) galaxies 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 PSB galaxies 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 PSB galaxies: central PSB (cPSB) galaxies with central quenching regions and ring PSB (rPSB) galaxies with quenching in their outskirts. We analyze a sample of 13 nearby (z < 0.1) PSB galaxies with spatially resolved optical IFS data from the Mapping Nearby Galaxies at Apache Point Observatory survey and matched resolution Atacama Large Millimeter/submillimeter Array observations of ¹²CO(1–0). Disturbed stellar kinematics in 7/13 of our PSB galaxies 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 inward and suppress star formation, which may include outflows, stellar bars, and minor mergers or interactions. The star formation efficiencies of the PSB regions in nearly half our galaxies are suppressed while the gas fractions are consistent with star-forming galaxies. Active galactic nucleus (AGN) feedback may drive this stabilization, and we observe AGN-consistent emission in the centers of 5/13 galaxies. Finally, our cPSB and rPSB galaxies have similar properties except the ionized and molecular gas in cPSB galaxies is more disturbed. Overall, the molecular gas in our PSB galaxies tends to be compact and highly disturbed, resulting in concentrated gas reservoirs unable to form stars efficiently.