We investigate the radial velocity (RV) variability and spectroscopic binarity of 19 Galactic long-period ($P_{rm{puls}} gtrsim 10$ d) classical Cepheid variable stars whose trigonometric parallaxes are being measured using the Hubble Space Telescope and Gaia. Our primary objective is to constrain possible parallax error due to undetected orbital motion. Using $>1600$ high-precision RVs measured between 2011 and 2016, we find no indication of orbital motion on $lesssim 5$ yr timescales for 18 Cepheids and determine upper limits on allowed configurations for a range of input orbital periods. The results constrain the unsigned parallax error due to orbital motion to $< 2 %$ for 16 stars, and $< 4 %$ for 18. We improve the orbital solution of the known binary YZ Carinae and show that the astrometric model must take into account orbital motion to avoid significant error ($sim pm 100 mu$arcsec). We further investigate long-timescale ($P_{rm{orb}} > 10$ yr) variations in pulsation-averaged velocity $v_gamma$ via a template fitting approach using both new and literature RVs. We discover the spectroscopic binarity of XZ Car and CD Cyg, find first tentative evidence for AQ Car, and reveal KN Cens orbital signature. Further (mostly tentative) evidence of time-variable $v_gamma$ is found for SS CMa, VY Car, SZ Cyg, and X Pup. We briefly discuss considerations regarding a vetting process of Galactic Leavitt law calibrators and show that light contributions by companions are insignificant for most distance scale applications.
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