Coastal erosion poses a major threat in the context of global environmental change and is influenced by both natural and anthropogenic processes. Traditional shallow water models often neglect the mechanical response of the seabed, limiting their predictive capacity in simulating sediment redistribution and shoreline evolution. This study extends the classical one-dimensional Saint-Venant shallow water equations by incorporating sediment transport, bottom friction, wave dispersion, and viscoelastic seabed behavior. The latter is modeled using the Kelvin–Voigt constitutive relation, which captures both elastic deformation and time-dependent viscous damping. The coupled system is implemented in the COMSOL Multiphysics platform as a set of partial differential equations. Theoretical case studies with different bathymetric configurations (steps or depressions on the seabed) are simulated to assess the influence of friction, dispersion, and seabed rheology. The numerical simulations highlight the stabilizing effects of viscoelastic behavior, especially when combined with dissipative mechanisms such as friction and dispersion. While the classical Saint-Venant model reproduces basic hydrodynamic responses, the inclusion of rheological terms leads to smoother water surface profiles and more realistic sediment redistribution. Notably, cases with viscoelastic seabeds exhibit damped morphological evolution, with reduced local instabilities and better agreement with observed erosion–deposition patterns in natural systems. The results demonstrate that incorporating viscoelastic properties into morphodynamic models improves the physical realism of simulations and enhances predictive capabilities. These findings support the integration of rheological behavior in coastal modeling frameworks, with potential applications in sediment management, risk assessment, and nature-based coastal defense design.