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The molecular gas in galaxies traces both the fuel for star formation and the processes that can enhance or suppress star formation. Observations of the molecular gas state can thus point to when and why galaxies stop forming stars. In this study, we present Atacama Large Millimeter/submillimeter Array observations of the molecular gas in galaxies evolving through the post-starburst phase. These galaxies have low current star formation rates (SFRs), regardless of the SFR tracer used, with recent starbursts ending within the last 600 Myr. We present CO (3–2) observations for three post-starburst galaxies, and dense gas HCN/HCO⁺/HNC (1–0) observations for six (four new) post-starburst galaxies. The post-starbursts have low excitation traced by the CO spectral-line energy distribution up to CO (3–2), more similar to early-type than starburst galaxies. The low excitation indicates that lower density rather than high temperatures may suppress star formation during the post-starburst phase. One galaxy displays a blueshifted outflow traced by CO (3–2). MaNGA observations show that the ionized gas velocity is disturbed relative to the stellar velocity field, with a blueshifted component aligned with the molecular gas outflow, suggestive of a multiphase outflow. Low ratios of HCO⁺/CO, indicating low fractions of dense molecular gas relative to the total molecular gas, are seen throughout post-starburst phase, except for the youngest post-starburst galaxy considered here. These observations indicate that the impact of any feedback or quenching processes may be limited to low excitation and weak outflows in the cold molecular gas during the post-starburst phase.