We present analyses of near-infrared spectroscopic data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) survey for 45 previously confirmed or candidate white dwarf-main-sequence (WDMS) binaries identified by the optical Sloan Digital Sky Survey (SDSS) and LAMOST surveys. Among these 45 systems, we classify three as having red giant primaries in the LAMOST sample and 14 as young stellar object contaminants in the photometrically identified SDSS sample. From among the subsample of 28 systems that we confirm to have MS primaries, we derive and place limits on orbital periods and velocity amplitudes for 14. Seven systems have significant velocity variations that warrant a post-common-envelope (PCE) binary classification, four of which are newly classified, three of which are newly confirmed, and five for which we can derive full orbital parameters. If confirmed, one of these newly discovered systems (2M14544500+4626456) will have the second-longest orbital period reported for a typical compact PCE WDMS binary (P = 15.1 days). In addition to the seven above, we also recover and characterize with APOGEE data the well-known PCE WDMS systems EG UMa and HZ 9. We also investigate the overall metallicity distribution of the WDMS sample, which is a parameter space not often explored for these systems. Of note, we find one system (2M14244053+4929580) to be extremely metal-poor ([Fe/H] = - 1.42) relative to the rest of the near-solar sample. Additionally, the PCE systems in our sample are found to be, on average, higher in metallicity than their wide-binary counterparts, though we caution that with this small number of systems, the sample may not be representative of the overall distribution of WDMS systems.