To date, few Earth System Models (ESMs) have had the ability to simulate the flow in ocean cavities below Antarctic ice shelves and its influence on ice-shelf basal melting. Yet capturing both this flow and the resulting melt patterns is critical for representing local, regional, and global feedbacks between the climate and sub-ice-shelf melting, as well as projecting future sea-level rise. Here, we present a small ensemble of historical simulations and SSP3-7.0 projections from the Energy Exascale Earth System Model (E3SM) version 2.1 that includes Antarctic ice-shelf cavities. The simulations have active ocean, sea-ice, atmosphere, land and river components. The model ocean domain has 12 km horizontal resolution around Antarctica, which is adequate for capturing dynamics in the larger ice-shelf cavities, regional aggregate melt fluxes, and water masses across most of the Antarctic continental shelf. The projections show significant warming and freshening of water masses on the Antarctic continental shelf, a deepening and poleward shift of the Amundsen Sea Low, and a significant increase in Antarctic ice-shelf melting through the 20th and 21st centuries. We also quantify the impact of evolving melt rates on ocean and climate conditions through the 21st century with a companion simulation where melt fluxes are kept fixed at 2015 levels. With this perturbation experiment we find local impacts on ocean conditions, for example in the Amundsen Sea Embayment, however global climate impacts are minimal. In addition to providing an estimate of future melting and other changes in regional and global climate under SSP3-7.0, these simulations are a stepping stone to coupled ice sheet-ocean simulations planned for the near future.