Several low-mass X-ray binaries display pronounced dips of variable depth and duration in their X-ray light curves, due to the periodic occultation of the central source by azimuthal accretion disk structure. We present results from Ginga observations of two such sources, XB 1916-053 and EXO 0748-676. XB 1916-053 was observed for 3 days in 1988 September. Periodicity studies show results consistent with those from previous Ginga observations (Smale et al. 1989), specifically that the derived X-ray period is shorter than the optical period by approximately 1 %. We discuss this problem in detail, drawing from current knowledge on CVs as well as LMXBs, and describe a more refined model for the system. If the accretion disk in XB 1916-053 is elliptical, the azimuthal position of the structure remains fixed in the binary reference frame, and thus the primary X-ray dips recur at the orbital period. However, the vertical extent of the disk structure, and thus the reprocessed optical emission, will be modulated on the beat between the orbital and disk precession periods, producing the observed period discrepancy. Three bursts were detected during the observation of XB 1916-053, the second of which occurred during a dip. Spectral analysis suggests that the second burst caused an almost instantaneous ionization of the absorbing medium in the line of sight; this material then returned to its equilibrium state on the same time scale as the burst decay. We show that the theoretical ionization and recombination time scales are short enough for this to be feasible, and that the total burst fluence is large enough to produce such an effect. In addition, we find evidence that the bursts cause a approximately threefold increase in the emission component from the disk. We observed EXO 0748 - 676 in 1989 March, also for 3 days. The source was found to be in a bright state, and data were obtained covering two eclipses and many dips. We have analyzed intensity-selected spectra accumulated from the dipping intervals of both XB 1916-053 and EXO 0748-676, and in addition have performed simulations to assess the effects of a rapidly changing column density upon the observed spectra from these sources. We show that during dipping intervals, where N(H) varies on a time scale shorter than the spectral integration time, spectral mixing can produce an apparent low-energy component in the data. Thus, rapidly variable simple absorption is indistinguishable from partial covering. From the variability analysis, we place loose limits upon the possible clumpiness of the material responsible for the accretion disk structure in XB 1916 - 053 and EXO 0748 - 676.