In this paper, a fully nonlinear Boussinesq model is presented and applied to the description of breaking waves and shoreline motions. It is based on Serre Green-Naghdi equations, solved using a time-splitting approach separating hyperbolic and dispersive parts of the equations. The hyperbolic part of the equations is solved using Finite-Volume schemes, whereas dispersive terms are solved using a Finite-Difference method. The idea is to switch locally in space and time to NSWE by skipping the dispersive step when the wave is ready to break, so as the energy dissipation due to wave breaking is predicted by the shock theory. This approach allows wave breaking to be handled naturally, without any ad-hoc parameterization for the energy dissipation. Extensive validations of the method are presented using laboratory data.