The characterization and mitigation of thermoacoustic combustion instabilities in gas turbine engines is necessary to reduce pollutant emissions, premature wear, and component failure associated with unstable flames. Fuel staging, a technique in which the fuel flow to a multi-nozzle combustor is unevenly distributed between the nozzles, has been shown to mitigate the intensity of self-excited combustion instabilities in multiple nozzle combustors. In our previous work, we hypothesized that staging suppresses instability through a phase-cancellation effect in which the heat release rate from the staged nozzle oscillates out of phase with that of the other nozzles, leading to destructive interference that suppresses the instability. This previous theory, however, was based on chemiluminescence imaging, which is a line-of-sight integrated technique. In this work, we use high-speed laser-induced fluorescence to further investigate instability suppression in two staging configurations: center-nozzle and outer-nozzle staging. An edge-tracking algorithm is used to compute local flame edge displacement as a function of time, allowing instability-driven edge oscillation phase coherence and other instantaneous flame dynamics to be spectrally and spatially resolved. Analysis of flame edge oscillations shows the presence of convecting coherent fluctuations of the flame edge caused by periodic vortex shedding. When the system is unstable, these two flame edges oscillate together as a result of high-intensity longitudinal-mode acoustic oscillations in the combustor that drive periodic vortex shedding at each of the nozzle exits. In the stable cases, however, the phase between the oscillations of the center and outer flame edges is greater than 90 degrees (~114 degrees), suggesting that the phase-cancellation hypothesis may be valid. This analysis allows a better understanding of the instantaneous flame dynamics behind flame edge oscillation phase offset and fuel staging-based instability suppression.