This work analyzes the recently introduced nonlinear dispersive shallow water wave equation, the Dullin-Gottwald-Holm (DGH) model. By taking into account a wider range of wave events that frequently occur in real shallow water systems, such as dispersive shocks, bores, and solitary waves, the advanced nonlinear dispersive shallow water model has been devised to predominate the classical models. This is accomplished by combining linear and nonlinear dispersion. The several traveling wave solutions of the governing model are systematically arranged using several well-known analytical methods. The main focus is to investigate how several important parameters affect the dynamics and shape of the resulting wave formations. As a result, many wave solutions, including solitary, shock, singular, shock-singular, hyperbolic, singular periodic, and rational waves, can be constructed by symbolic computations. The parameter regimes guarantee that a careful analysis determines the generated solutions' validity and physical significance. The interpretation of the results is improved by displaying the solutions across a variety of parameter values using contour graphs, 2D plots, and 3D surface representations. These results confirm the efficiency and dependability of the used computational paradigm, which has a great deal of promise for further studies looking at more intricate nonlinear processes.