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We present Magellan/IMACS, Anglo-Australian Telescope/AAOmega+2dF, and Very Large Telescope/GIRAFFE+FLAMES spectroscopy of the Carina II (Car II) and Carina III (Car III) dwarf galaxy candidates, recently discovered in the Magellanic Satellites Survey (MagLiteS). We identify 18 member stars in Car II, including two binaries with variable radial velocities and two RR Lyrae stars. The other 14 members have a mean heliocentric velocity and a velocity dispersion of . Assuming Car II is in dynamical equilibrium, we derive a total mass within the half-light radius of , indicating a mass-to-light ratio of /. From equivalent width measurements of the calcium triplet lines of nine red giant branch (RGB) stars, we derive a mean metallicity of with dispersion . Considering both the kinematic and chemical properties, we conclude that Car II is a dark-matter-dominated dwarf galaxy. For Car III, we identify four member stars, from which we calculate a systemic velocity of . The brightest RGB member of Car III has a metallicity of . Due to the small size of the Car III spectroscopic sample, we cannot conclusively determine its nature. Although these two systems have the smallest known physical separation () among Local Group satellites, the large difference in their systemic velocities, , indicates that they are unlikely to be a bound pair. One or both systems are likely associated with the Large Magellanic Cloud (LMC), and may remain LMC satellites today. No statistically significant excess of γ-ray emission is found at the locations of Car II and Car III in eight years of Fermi-LAT data. [ABSTRACT FROM AUTHOR]