Quasars whose broad emission lines show temporal, bulk radial velocity (RV) shifts have been proposed as candidate sub-parsec (sub-pc), binary supermassive black holes (BSBHs). We identified a sample of 16 BSBH candidates based on two-epoch spectroscopy among 52 quasars with significant RV shifts over a few rest-frame years. The candidates showed consistent velocity shifts independently measured from two broad lines (H$beta$ and H$alpha$ or Mg${rm ,II}$) without significant changes in the broad-line profiles. Here in the third paper of the series, we present further third- and fourth-epoch spectroscopy for 12 of the 16 candidates for continued RV tests, spanning $sim$5--15 yr in the quasars rest frames. Cross-correlation analysis of the broad H$beta$ calibrated against [O${rm,III}],lambda 5007$ suggests that 5 of the 12 quasars remain valid as BSBH candidates. They show broad H$beta$ RV curves that are consistent with binary orbital motion without significant changes in the broad line profiles. Their broad H$alpha$ (or Mg${rm ,II}$) lines display RV shifts that are either consistent with or smaller than those seen in broad H$beta$. The RV shifts can be explained by a $sim$0.05--0.1 pc BSBH with an orbital period of $sim$40--130 yr, assuming a mass ratio of 0.5--2 and a circular orbit. However, the parameters are not well constrained given the few epochs that sample only a small portion of the hypothesized binary orbital cycle. The apparent occurrence rate of sub-pc BSBHs is $lesssim$13$pm$5% among all SDSS quasars, with no significant difference in the subsets with and without single-epoch broad line velocity offsets. Dedicated long-term spectroscopic monitoring is still needed to further confirm or reject these BSBH candidates.
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