We present CO(1-0) maps of 12 warm H$_2$-selected Hickson Compact Groups (HCGs), covering 14 individually imaged warm H$_2$ bright galaxies, with CARMA. We found a variety of molecular gas distributions within the HCGs, including regularly rotating disks, bars, rings, tidal tails, and possibly nuclear outflows, though the molecular gas morphologies are more consistent with spirals and early-type galaxies than mergers and interacting systems. Our CO-imaged HCG galaxies show star formation suppression of $langle$S$rangle$=10$pm$5, distributed bimodally, with five objects exhibiting suppressions of S$gtrsim$10 and depletion timescales $gtrsim$10Gyr. This star formation inefficiency is also seen in the efficiency per freefall time. We investigate the gas-to-dust ratios of these galaxies to determine if an incorrect conversion caused the apparent suppression and find that HCGs have normal ratios. It is likely that the cause of the suppression in these objects is associated with shocks injecting turbulence into the molecular gas. Galaxies with high star formation suppression (S$gtrsim$10) also appear to be those in the most advanced stages of transition across optical and infrared color space. This supports the idea that some galaxies in HCGs are transitioning objects, where a disruption of the existing molecular gas in the system suppresses star formation by inhibiting the molecular gas from collapsing and forming stars efficiently. These observations, combined with recent work on poststarburst galaxies with molecular reservoirs, indicates that galaxies do not need to expel their molecular reservoirs prior to quenching star formation and transitioning from blue spirals to red early-type galaxies. This may imply that star formation quenching can occur without the need to starve a galaxy of cold gas first.
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