This study investigates the time–fractional Benjamin–Bona–Mahony (BBM) equation and the time–space fractional Benjamin–Bona–Mahony–Burgers (BBMB) equation, which model nonlinear wave propagation and dispersive transport phenomena in complex media exhibiting memory and nonlocal effects. To obtain efficient analytical approximations, the Shehu Transform Adomian Decomposition Method (STADM), a novel hybrid framework that combines the Shehu Transform (ST) with the Adomian Decomposition Method (ADM), is employed to derive series-form analytical solutions for both models. A four-term series approximation is obtained for each equation, and its convergence is verified through direct comparison with the corresponding exact analytical solution. The results exhibit excellent agreement, particularly as the fractional order approaches unity ($\alpha \to 1$), confirming the reliability and high accuracy of the proposed method. The influence of time and space fractional orders on the dynamic behavior of the solutions is analyzed through comprehensive 3D plots, which illustrate the effect of fractional parameters on wave dispersion and propagation. The main contributions of this work are: (i) the first systematic application of the STADM framework to the time–space fractional BBM and BBMB equations; (ii) the derivation of rapidly convergent analytical series solutions validated against exact results; and (iii) the demonstration of how fractional parameters modulate physical wave and diffusion processes. The proposed approach provides a simple, accurate, and computationally efficient tool for solving nonlinear fractional models relevant to wave evolution, pollutant dispersion, and oil-spill diffusion in coastal and environmental systems.