Semiconductor nanocrystals or quantum dots (QDs) are now widely used across solar cell, display, and bioimaging technologies. While advances in multishell, alloyed, and multinary core-shell QD structures have led to improved light-harvesting and photoluminescence (PL) properties of these nanomaterials, the effects that QD-capping have on the exciton dynamics that govern PL instabilities such as blinking in single-QDs is not well understood. We report experimental measurements of shell-size-dependent absorption and PL intermittency in CdSe-CdS QDs that are consistent with a modified charge-tunnelling, self-trapping (CTST) description of the exciton dynamics in these nanocrystals. By introducing an effective, core-exciton size, which accounts for delocalization of charge carriers across the QD core and shell, we show that the CTST models both the shell-depth-dependent red-shift of the QD band gap and changes in the on/off-state switching statistics that we observe in single-QD PL intensity trajectories. Further analysis of CdSe-ZnS QDs, shows how differences in shell structure and integrity affect the QD band gap and PL blinking within the CTST framework.