We present high-resolution spectra and spectroastrometric (SA) measurements of fundamental rovibrational CO emission from nine nearby (≲300 pc) protoplanetary disks where large inner dust cavities have been observed. The emission-line profiles and SA signals are fit with a slab disk model that allows the eccentricity of the disk and intensity of the emission to vary as power laws. Six of the sources are well fit with our model, and three of these sources show asymmetric line profiles that can be fit by adopting a nonzero eccentricity. The three other sources have components in either their line profile or SA signal that are not captured by our disk model. Two of these sources (V892 Tau and CQ Tau) have multi-epoch observations that reveal significant variability. CQ Tau and AB Aur have CO line profiles with centrally peaked components that are similar to line profiles which have been interpreted as evidence of molecular gas arising from a wide-angle disk wind. Alternatively, emission from a circumplanetary disk could also account for this component. The interpretations of these results can be clarified in the future with additional epochs that will test the variability timescale of these SA signals. We discuss the utility of using high-resolution spectroscopy for probing the dynamics of gas in the disk and the scenarios that can give rise to profiles that are not fit with a simple disk model.