Dynamical behaviors of the time-fractional nonlinear model of the coupled spring-mass system with damping have been explored here. Fractional derivatives with singular and non-singular kernels are used to assess the suggested model. The fractional Adams-Bashforth numerical method based on Lagrange polynomial interpolation is applied to solve the system with non-local operators. Existence, Ulam-Hyers stability, and uniqueness of the solution are established by using fixed-point theory and nonlinear analysis. Further, the error analysis of the present method has also been included. Finally, the behavior of the solution is explained by graphical representations through numerical simulations.