This work addressed one of the most essential evolutionary equations that was widely used in describing various nonlinear wave propagation and dispersive phenomena in various scientific and engineering applications, which was called the nonlinear fractional Fornberg-Whitham (FFW). Due to the importance of this equation, we examined it by employing two highly effective techniques: the residual power series method (RPSM) and the new iterative approach (NIM), both distinguished by their efficacy in solving more complicated nonlinear fractional evolutionary equations. Moreover, we integrated the Elzaki transform with both approaches to create Elzaki RPSM (ERPSM) and Elzaki NIM (ENIM) to ease the calculations. The ERPSM effectively combined the power series approach with residual error analysis to generate highly accurate series solutions, while ENIM provided alternative frameworks for handling nonlinearities and achieving rapid convergence. Comparative studies of the obtained solutions highlighted these methods' efficiency, accuracy, and reliability in solving fractional-order differential equations. The results underscored the potential of these analytical techniques for modeling and solving complex fractional wave equations, contributing to the advancement of mathematical physics and computational fluid dynamics.