Recent detections of a significant interstellar medium (ISM) in many post-starburst galaxies challenges their traditional classification as dust- and gas-poor merger remnants, rapidly transitioning to quiescence. We have conducted a multi-wavelength survey of 33 E +A post-starbursts selected from SDSS, including Spitzer, Herschel, and ALMA. We find compact, warm dust reservoirs with high polycyclic aromatic hydrocarbon (PAH) abundances. Their infrared spectral properties are unique, with dominant PAH emission, very weak nebular lines, and deep [C II] deficits – the aging burst populations provide an unusual “high-soft” radiation field that seemingly dominates the post- starbursts’ ISM energetics. We also find unusually strong H2 rotational emission, among the brightest, relative to infrared luminosity, of any known galaxy – indicating a highly turbulent ISM. High-resolution ALMA CO(2–1) follow-up observations reveal that the molecular gas in these post-starbursts is exceptionally dense – rivaling the gas found in ULIRGs – and is concentrated in turbulent, rotating nuclear reservoirs. Coupled with modest SFRs from a range of infrared tracers, the post-starbursts’ high gas densities indicate highly inefficient star formation – they lie more than an order of magnitude below the Kennicutt-Schmidt star-forming relation. These results paint a coherent picture of systems in which star formation was, indeed, rapidly truncated, but in which the ISM was not completely expelled. High-density nuclear reservoirs of molecular fuel remain but are supported against collapse by some form of continual turbulent heating.