Future thermosteric and dynamic sea-level changes are often projected by process-based climate models. Emulation of such computationally expensive models helps enable model intercomparison over a range of forcing scenarios and thus enables additional analysis of sea-level rise projection uncertainty. Current emulation methods use linear response functions to estimate global mean sea-level response. Here, we introduce a novel dual model to emulate global mean thermosteric sea-level rise that incorporates short- and long-term responses to climate forcing. This nonlinear response function outperforms existing linear response functions over six illustrative general circulation models and the four representative concentration pathways. To emulate dynamic sea-level projections, we introduce a linear pattern scaling model that relates regional sea-level changes to global mean thermosteric sea-level rise. Pattern scaling is shown to reproduce strongly forced sea-level trends. Our results demonstrate effective emulation of global and regional sea-level rise, which can facilitate the consideration of sea-level rise projection uncertainty critical to the analysis of sea-level rise hazard.