In coastal areas, bathymetric data are nowadays available at resolutions significantly higher than those of the computational mesh. This disparity poses several challenges for high-order finite element discretizations of the shallow water equations: the mesh may not be aligned with strong bathymetric gradients, and the finite element method must be able to represent such gradients, as well as sub-grid bathymetric features, within individual elements. The choice of evolving the water depth, although straightforwardly mass-conserving, is only justified under strong smoothness assumptions for the bathymetry over the element. Real bathymetries are typically irregular, which is reflected in a lack of regularity of the water depth, also in those elements for which the water depth is zero at some location, as a result of localized drying. Therefore limiters or smoothing techniques must be activated on the bathymetry, which is unfortunate for very accurate bathymetric datasets. In [V. Casulli, International Journal of Numerical Methods in Fluids, Vol. 60, 2009.], a finite difference method that employs the free-surface as prognostic variable and fully exploits high-resolution bathymetric datasets has been proposed, including a rigorous and mass-conserving formulation of wetting and drying with a sub-grid irregular bathymetry. The purpose of this work is to present a high-order adaptive finite element solver for the shallow water equations, which is also capable of exploiting high resolution bathymetries. We establish several properties of the resulting numerical scheme with irregular bathymetry: well-balancing, water depth positivity under a CFL condition that depends on the accuracy of the quadrature formula, i.e., the accuracy of the sub-grid bathymetry and a consistent discretization of passive tracers with the continuity equation. The spatial discretization is based on a high-order Discontinuous Galerkin (DG) method, as implemented in the deal.II library. In this framework, we test the robustness of the proposed method also with dynamic Adaptive Mesh Refinement to simulate the tidal circulations in complex coastal environments.