We use observations from the Hubble Space Telescope Advanced Camera for Surveys (HST/ACS) study of Galactic globular clusters to investigate the spatial distribution of the inner regions of the disrupting Sagittarius dwarf spheroidal galaxy (Sgr). We combine previously published analyses of four Sgr member clusters located near or in the Sgr core (M54, Arp 2, Terzan 7, and Terzan 8) with a new analysis of diffuse Sgr material identified in the background of five low-latitude Galactic bulge clusters (NGC 6624, 6637, 6652, 6681, and 6809) observed as part of the ACS survey. By comparing the bulge cluster color-magnitude diagrams to our previous analysis of the M54/Sgr core, we estimate distances to these background features. The combined data from four Sgr member clusters and five Sgr background features provide nine independent measures of the Sgr distance and, as a group, provide uniformly measured and calibrated probes of different parts of the inner regions of Sgr spanning 20° over the face of the disrupting dwarf. This allows us, for the first time, to constrain the three-dimensional orientation of Sgr's disrupting core and globular cluster system and compare that orientation to the predictions of an N-body model of tidal disruption. The density and distance of Sgr debris are consistent with models that favor a relatively high Sgr core mass and a slightly greater distance (28-30 kpc, with a mean of 29.4 kpc). Our analysis also suggests that M54 is in the foreground of Sgr by ~2 kpc, projected on the center of the Sgr dSph. While this would imply a remarkable alignment of the cluster and the Sgr nucleus along the line of sight, we cannot identify any systematic effect in our analysis that would falsely create the measured 2 kpc separation. Finally, we find that the cluster Terzan 7 has the most discrepant distance (25 kpc) among the four Sgr core clusters, which may suggest a different dynamical history than the other Sgr core clusters.