We have used the Atacama Large Millimeter/submillimeter Array (ALMA) to carry out a search for CO (3$-$2) or (4$-$3) emission from the fields of 12 high-metallicity ([M/H]~$geq -0.72$,dex) damped Lyman-$alpha$ absorbers (DLAs) at $z approx 1.7-2.6$. We detected CO emission from galaxies in the fields of five DLAs (two of which have been reported earlier), obtaining high molecular gas masses, $rm M_{mol} approx (1.3 - 20.7) times (alpha_{rm CO}/4.36) times 10^{10} ; M_odot$. The impact parameters of the CO emitters to the QSO sightline lie in the range $b approx 5.6-100$~kpc, with the three new CO detections having $b lesssim 15$~kpc. The highest CO line luminosities and inferred molecular gas masses are associated with the highest-metallicity DLAs, with [M/H]~$gtrsim -0.3$,dex. The high inferred molecular gas masses may be explained by a combination of a stellar mass-metallicity relation and a high molecular gas-to-stars mass ratio in high-redshift galaxies; the DLA galaxies identified by our CO searches have properties consistent with those of emission-selected samples. None of the DLA galaxies detected in CO emission were identified in earlier optical or near-IR searches and vice-versa; DLA galaxies earlier identified in optical/near-IR searches were not detected in CO emission. The high ALMA CO and C[{sc ii}]~158$mu$m detection rate in high-$z$, high-metallicity DLA galaxies has revolutionized the field, allowing the identification of dusty, massive galaxies associated with high-$z$ DLAs. The H{sc i}-absorption criterion identifying DLAs selects the entire high-$z$ galaxy population, including dusty and UV-bright galaxies, in a wide range of environments.
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