We have been spectroscopically monitoring 88 quasars selected to have broad H$beta$ emission lines offset from their systemic redshift by thousands of km s$^{-1}$. By analogy with single-lined spectroscopic binary stars, we consider these quasars to be candidates for hosting supermassive black hole binaries (SBHBs). In this work we present new radial velocity measurements, typically 3-4 per object over a time period of up to 12 years in the observers frame. In 29/88 of the SBHB candidates no variability of the shape of the broad H$beta$ profile is observed, which allows us to make reliable measurements of radial velocity changes. Among these, we identify three objects that have displayed systematic and monotonic velocity changes by several hundred km s$^{-1}$ and are prime targets for further monitoring. Because the periods of the hypothetical binaries are expected to be long, we cannot hope to observe many orbital cycles during our lifetimes. Instead, we seek to evaluate the credentials of the SBHB candidates by attempting to rule out the SBHB hypothesis. In this spirit, we present a method for placing a lower limit on the period, and thus the mass, of the SBHBs under the assumption that the velocity changes we observe are due to orbital motion. Given the duration of our monitoring campaign and the uncertainties in the radial velocities, we were able to place a lower limit on the total mass in the range $4.7times10^4-3.8times10^8$ $M_{scriptscriptstyle odot}$, which does not yet allow us to rule out the SBHB hypothesis for any candidates.
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