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We present Infrared Space Observatory spectroscopy of sites in the supernova remnants W28, W44, and 3C 391, where blast waves are impacting molecular clouds. The complete wavelength range from 42 to 188 µm was observed with the Long Wavelength Spectrometer, as well as narrow ranges centered on 4.695, 9.665, 25.98, and 34.82 µm with the Short Wavelength Spectrometer. Atomic fine-structure lines were detected from (in order of atomic number): C⁺, N⁺, N⁺⁺, O⁰, O⁺⁺, O⁺⁺⁺, Si⁺, P⁺, and Fe⁺. The two lines of H₂ that we observed, S(3) and S(9), were detected for all three remnants. The observations require both shocks into gas with moderate (∼10² cm^–3) and high (∼10⁴ cm^–3) preshock densities, with the moderate-density shocks producing the ionic lines and the high-density shock producing the molecular lines. No single shock model can account for all of the observed lines, even at the order of magnitude level. We find that the principal coolants of radiative supernova shocks in moderate-density gas are the far-infrared continuum from dust grains surviving the shock, followed by collisionally excited [O I] 63.2 µm and [Si II] 34.8 µm lines. The principal coolant of the high-density shocks is collisionally excited H₂ rotational and ro-vibrational line emission. We systematically examine the ground-state fine structure of all cosmically abundant elements to explain the presence or lack of all atomic fine lines in our spectra in terms of the atomic structure, interstellar abundances, and a moderate-density, partially ionized plasma. The [P II] line at 60.6 µm is the first known astronomical detection, but its brightness can be explained using the solar abundance of P. There is only one, bright unidentified line in our spectra, at 74.26 µm; as there is no plausible atomic fine-structure line at this wavelength, we suggest this line is molecular. The presence of bright [Si II] and [Fe II] lines requires partial destruction of the dust. The required gas-phase abundance of Fe suggests 15%-30% of the Fe-bearing grains were destroyed. Adding the Si and Fe gas mass, and correcting for the mass of other elements normally found in dust, we find ∼0.5 M_☉ of dust vapors from the shocked clump 3C 391:BML. The infrared continuum brightness requires ∼1 M_☉ of dust survives the shock, suggesting about 1/3 of the dust mass was destroyed, in agreement with the depletion estimate and with theoretical models for dust destruction.