We present the results of a precise near-infrared (NIR) radial velocity (RV) survey of 32 low-mass stars with spectral types K2-M4 using CSHELL at the NASA IRTF in the $K$-band with an isotopologue methane gas cell to achieve wavelength calibration and a novel iterative RV extraction method. We surveyed 14 members of young ($approx$ 25-150 Myr) moving groups, the young field star $varepsilon$ Eridani as well as 18 nearby ($<$ 25 pc) low-mass stars and achieved typical single-measurement precisions of 8-15 m s$^{-1}$ with a long-term stability of 15-50 m s$^{-1}$. We obtain the best NIR RV constraints to date on 27 targets in our sample, 19 of which were never followed by high-precision RV surveys. Our results indicate that very active stars can display long-term RV variations as low as $sim$ 25-50 m s$^{-1}$ at $approx$ 2.3125 $mu$m, thus constraining the effect of jitter at these wavelengths. We provide the first multi-wavelength confirmation of GJ 876 bc and independently retrieve orbital parameters consistent with previous studies. We recovered RV variability for HD 160934 AB and GJ 725 AB that are consistent with their known binary orbits, and nine other targets are candidate RV variables with a statistical significance of 3-5$sigma$. Our method combined with the new iSHELL spectrograph will yield long-term RV precisions of $lesssim$ 5 m s$^{-1}$ in the NIR, which will allow the detection of Super-Earths near the habitable zone of mid-M dwarfs.
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