This manuscript focuses on nonlinear wave dynamics in shallow water, namely the improved Boussinesq equation. The initial and boundary value problem for the aforementioned equation is addressed, incorporating both weak and strong damping terms. The primary focus of this study is to explore the continuous dependence of solutions on the coefficients associated with these damping terms. By investigating this dependence, we uncover important insights into the sensitivity of the solutions to variations in the damping parameters. Our results reveal how these coefficients influence the qualitative behavior of the solutions, including their stability and long-term dynamics. In conclusion, we provide a detailed discussion of the implications of the damping coefficients, highlighting their significant impact on the solution structure. The findings are also contrasted with existing results in the literature, offering new perspectives on the interplay between damping effects and solution behavior. These contributions are expected to be of considerable interest to researchers studying the improved Boussinesq equation and related models in the context of wave dynamics and dissipative systems.