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Short-period cataclysmic variables (spCVs), with orbital periods below the period gap ($P_{orb}$ < 2 hr), offer insight into the evolutionary models of CVs and can serve as strong emitters of gravitational waves (GWs). To identify new spCV candidates, we crossmatch a catalog of known CVs to sources with robust parallaxes in the Gaia second data release (DR2). We uncover and fit an apparently monotonic relationship between the color--absolute-magnitude diagram (CMD) position and $P_{orb}$ of these CVs, revealed in DR2. To supplement this relation, we develop a method for identifying sources with large photometric variability, a characteristic trait of spCVs. Using all available Gaia light curves, we construct a machine-learned regression model to predict variability metrics for sources in the CMD locus of known spCVs based solely on time-averaged covariates present in DR2. Using this approach we identify 3,253 candidate spCVs, of which $\sim$95% are previously unknown. Inspection of archival SDSS spectra of these candidates suggests that $>$82% are likely to be spCVs: a noticeably higher recovery rate than previous light curve searches, which bias toward active systems. We obtain optical spectra of 9 new systems at Lick Observatory and confirm that all objects are CV systems. We measure $P_{orb}$ for 7 systems using archival Gaia and Palomar Transient Factory light curves, 3 of which do not have previous $P_{orb}$ measurements. We use the CMD-$P_{orb}$ relation to infer the detectability of these systems to the upcoming LISA mission, and find that six sources may be coherent LISA verification binaries, with an estimated SNR > 5 in the 4 yr mission. This paper demonstrates that the time-averaged Gaia catalog is a powerful tool in the methodical discovery and characterization of time-varying objects, making it complementary to missions like ZTF, TESS, and the Vera Rubin LSST.