Charged exciton states commonly occur both in spectroscopic studies of quantum dots (QDs) and during operation of QD-based devices. The extra charge added to the neutral exciton modifies its radiative decay rate and also opens an additional nonradiative pathway associated with an Auger process whereby the recombination energy of an exciton is transferred to the excess charge. Here we conduct single-dot spectroscopic studies of Auger recombination in thick-shell ("giant") CdSe/CdS QDs with and without an interfacial alloy layer using time-tagged, time-correlated single-photon counting. In photoluminescence (PL) intensity trajectories of some of the dots, we resolve three distinct states of different emissivities ("bright", "gray", and "dark") attributed, respectively, to the neutral exciton and negative and positive trions. Simultaneously acquired PL lifetime trajectories indicate that the positive trion is much shorter lived than the negative trion, which can be explained by a high density of valence band states and a small hole localization radius (defined by the QD core size), factors that favor an Auger process involving intraband excitation of a hole. A comparison of trion and biexciton lifetimes suggests that the biexciton Auger decay can be treated in terms of a superposition of two independent channels associated with positive- and negative-trion pathways. The resulting interdependence between Auger time constants might simplify the studies of multicarrier recombination by allowing one, for example, to infer Auger lifetimes of trions of one sign based on the measurements of biexciton decay and dynamics of the trions of the opposite sign or, alternatively, estimate the biexciton lifetime based on studies of trion dynamics.