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An understanding of the mass build-up in galaxies over time necessitates tracing the evolution of cold gas (molecular and atomic) in galaxies. To that end, we have conducted a pilot study called CO Observations with the LMT of the Blind Ultra-Deep H I Environment Survey (COOL BUDHIES).We have observed 23 galaxies in and around the two clusters Abell 2192 (z = 0.188) and Abell 963 (z = 0.206), where 12 are cluster members and 11 are slightly in the foreground or background, using about 28 total hours on the Redshift Search Receiver on the Large Millimeter Telescope (LMT) to measure the 12CO J= 1→0 emission line and obtain molecular gas masses. These new observations provide a unique opportunity to probe both the molecular and atomic components of galaxies as a function of environment beyond the local Universe. For our sample of 23 galaxies, nine have reliable detections (S/N ≥ 3.6) of the 12COline, and another six have marginal detections (2.0 < S/N < 3.6). For the remaining eight targets we can place upper limits on molecular gas masses roughly between 109 and 1010M☉. Comparing our results to other studies of molecular gas, we find that our sample is significantly more abundant in molecular gas overall, when compared to the stellar and the atomic gas component, and our median molecular gas fraction lies about 1σ above the upper limits of proposed redshift evolution in earlier studies. We discuss possible reasons for this discrepancy, with the most likely conclusion being target selection and Eddington bias. [ABSTRACT FROM AUTHOR]