This work discusses the dual-mode form of the nonlinear Schrödinger equation, which depicts the augmentation or absorption of dual waves. This model scrutinizes the contemporaneous generation of two characteristic waves dealing with dual-mode and introduces three physical parameters: nonlinearity, phase velocity, and dispersive factor. The wave phenomena of the obtained solutions are applied to water wave mechanics, fluid dynamics, ocean engineering, and science. We use two different methods for the dual-mode nonlinear Schrödinger equation (DMNLSE): the new extended direct algebraic method (NEDAM) and the dynamical system method (bifurcation analysis). These methods were not applied to the DMNLSE before. Firstly, by using the NEDAM, we observe that the DMNLSE has the shape of mixed-trigonometric solutions, shock solutions, singular solutions, complex dark-bright solutions, mixed-singular solutions, trigonometric solutions, different types of complex-combo solutions, periodic and mixed-periodic solutions, mixed-hyperbolic solutions, and a plane solution. Indeed, thanks to the NEDAM, the families of rational solutions have also appeared during the derivation. Secondly, the bifurcation analysis of the model in question is performed, and the fixed points are systematically generated. Thus, other solutions of various types are disclosed. To show the physical significance of the respected model, some three-dimensional and contour plot graphs of the obtained outcomes are illustrated with the help of Mathematica under the appropriate option concerning parameter values. We have made comparisons between our solution and other solutions in the previous literature. The obtained outcomes are beneficial to studying and corroborating the analytical solutions with numerical and experimental work in nonlinear dynamics modeled by the equation. Consequently, it revealed that the mentioned techniques can be a conceivable tool for creating unique precise soliton solutions for different needs, which play a paramount role in applied science and engineering.