We construct a simple morphodynamic model to investigate the long-term dynamic evolution of a coastal barrier system experiencing sea-level rise. Using a simplified barrier geometry, the model includes a dynamic shoreface profile that can be out of equilibrium and explicitly treats barrier sediment overwash as a flux. With barrier behavior primarily controlled by the maximum potential overwash flux and the rate of shoreface response, the modeled barrier system demonstrates four primary behaviors: height drowning, width drowning, constant landward retreat, and a periodic retreat. Height drowning occurs when overwash fluxes are insufficient to maintain the landward migration rate required to keep pace with sea-level rise. On the other hand, width drowning occurs when the shoreface response rate is insufficient to maintain the barrier geometry during overwash-driven landward migration. During periodic barrier retreat, the barrier experiences oscillating periods of rapid overwash followed by periods of relative stability as the shoreface resteepens. This periodic retreat, which occurs even with a constant sea-level rise rate, arises when overwash rates and shoreface response rates are large and of similar magnitude. We explore the occurrence of these behaviors across a wide range of parameter values and find that in addition to the maximum overwash flux and the shoreface response rate, barrier response can be particularly sensitive to the sea-level rise rate and back-barrier lagoon slope. Overall, our findings contrast with previous research which has primarily associated complex barrier behavior with changes in external forcing such as sea-level rise rate, sediment supply, or back-barrier geometry.