We present a search for new planetary-mass members of nearby young moving groups (YMGs) using astrometry for 694 T and Y dwarfs, including 447 objects with parallaxes, mostly produced by recent large parallax programs from UKIRT and Spitzer. Using the BANYAN $Sigma$ and LACEwING algorithms, we identify 30 new candidate YMG members, with spectral types of T0$-$T9 and distances of $10-43$ pc. Some candidates have unusually red colors and/or faint absolute magnitudes compared to field dwarfs with similar spectral types, providing supporting evidence for their youth, including 4 early-T dwarfs. We establish one of these, the variable T1.5 dwarf 2MASS J21392676$+$0220226, as a new planetary-mass member ($14.6^{+3.2}_{-1.6}$ M$_{rm Jup}$) of the Carina-Near group ($200pm50$ Myr) based on its full six-dimensional kinematics, including a new parallax measurement from CFHT. The high-amplitude variability of this object is suggestive of a young age, given the coexistence of variability and youth seen in previously known YMG T dwarfs. Our four latest-type (T8$-$T9) YMG candidates, WISE J031624.35$+$430709.1, ULAS J130217.21$+$130851.2, WISEPC J225540.74$-$311841.8, and WISE J233226.49$-$432510.6, if confirmed, will be the first free-floating planets ($approx2-6$ M$_{rm Jup}$) whose ages and luminosities are compatible with both hot-start and cold-start evolutionary models, and thus overlap the properties of the directly-imaged planet 51 Eri b. Several of our early/mid-T candidates have peculiar near-infrared spectra, indicative of heterogenous photospheres or unresolved binarity. Radial velocity measurements needed for final membership assessment for most of our candidates await upcoming 20$-$30 meter class telescopes. In addition, we compile all 15 known T7$-$Y1 benchmarks and derive a homogeneous set of their effective temperatures, surface gravities, radii, and masses.
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