Core/thick-shell giant quantum dots (gQDs) possessing type II electronic structures exhibit suppressed blinking and diminished nonradiative Anger recombination. We investigate CdSe/ZnSe and ZASe/CdS as potential new gQDs. We Show theoretically and experimentally that both can exhibit partial or complete spatial separation of an excited-state electron hole pair (i.e., type II behavior). However; we reveal that thick-shell growth is challenged by competing processes: alloying and cation exchange. We demonstrate that these can be largely avoided by choice of shelling conditions (e.g., time, temperature, and QD core identity). The resulting CdSe/ZnSe gQDs exhibit unusual single-QD properties, principally emitting from dim gray states but having high two-exciton (biexciton) emission efficiencies, whereas ZnSe/CdS gQDs show characteristic gQD blinking suppression, though only if shelling is accompanied by partial cation exchange.