Unless you develop/implement a method to track the wet/dry point, you need to look into the "precursor film" trick (the surface is slightly wetted by introducing a minimum film height all over the domain, say 10^{-6}).

How then can one implement that. I want to learn from first principle. But the papers I have are not presenting the fundamental steps. Please, kindly help me with some starting materials.

Note that the 2D-shallow water model excludes viscous effects; thus it inherently cannot describe the dynamics of dewetting that would lead to dry spots. Further, the shallow water model cannot describe the flow of thin liquid films near a boundary, the dynamics of which are dominated by viscous effects. I think the approach you will have to take is one based on a singular perturbation analysis, in which the outer solution is described by the shallow water wave model and the inner solution ( when the layer thickness is sufficiently thin) described by a lubrication-like flow dominated by viscosity and surface tension. There are many examples in the literature that discuss deweting of thin films using a lubrication model.

But your challenge will be to define the appropriate scalings such that you have the correct limits as the layer becomes vanishingly small in thickness, and have the correct matching conditions.

Perhaps others on the forum know if this has been done before- I am not familiar with the literature of shallow water waves....