High-resolution millimeter-wave (CO, CS, and HCO+ rotational lines) and near- infrared (H-2 2.12 mum rovibrational and Fe II fine-structure lines) observations of the supernova remnants W28 and W44 reveal extensive shocked molecular gas, where supernova blast waves are propagating into giant molecular clouds. New CO observations were carried out with the IRAM 30 m and Arizona Radio Observatory 12 m telescopes, and the near-infrared observations were with the PFIRCAM and Wide-Field Infrared Camera on the Palomar Hale 200 inch (5.08 m) telescope. The near- infrared observations reveal shocked H-2 emission from both supernova remnants, showing intricate networks of filaments on arcsecond scales, following the bright ridges of the radio shells. The emission is particularly bright in the northeastern, southern, and western parts of W44 and in the eastern bar in W28. The H-2 emission reveals some bright, clumpy structures, as well as very thin filamentary structures likely to be individual shock fronts seen edge-on. The high-resolution IRAM CO(2-->1) and CS(2-->1) spectra clearly distinguish between the shocked and preshock gas for most of the supernova remnants. Some of the CO spectra appear to have multiple components, but the less optically thick (CO)-C-13 lines clearly demonstrate that the CO(2-->1) lines are broad, with deep absorption dips caused by cold, dense gas in the line of sight. The CO and CS line widths, indicative of the shock speed, are 20-30 km s(-1). Both the near- infrared and millimeter-wave emission are attributed to shocks into gas with density higher than 10(3) cm(-3). Individual shock structures are resolved in the H-2 emission, with inferred edge-on shock thicknesses of similar to10(17) cm, consistent with nondissociative shocks into gas densities of 10(3)-10(4) cm(-3). Bright 1720 MHz OH masers are located within the shocked H-2 gas complexes and highlight only localized areas, where the conditions for masing are optimal. The Halpha and X-ray emission, which trace hotter shocked gas, have morphologies very different from the radio. We find a detailed correlation of the radio and H-2 emission for some long filaments, indicating cosmic-ray acceleration or reacceleration due to shocks into moderately dense gas. Compared to the interclump gas and the very dense cores, the synchrotron emissivity of the moderate-density (CO emitting) medium is highest, which explains the radio-H-2 correlation and the very bright radio emission of these two supernova remnants despite their relatively advanced age. The different morphologies of these two remnants at different wavelengths are explained by the highly nonuniform structure of giant molecular clouds, with low-density (similar to5 cm(-3)) gas occupying most (similar to90%) of the volume, moderate-density gas (similar to10(3) cm(-3)) gas occupying most of the rest of the volume, and dense gas in the cores.