Fish-inspired aquatic robots are gaining increasing attention in marine robot communities due to their high swimming speeds and efficient propulsion enabled by flexible bodies that generate undulatory motions. To support the design optimization and control of such systems, accurate, interpretable, and computationally tractable modeling of the underlying swimming dynamics is indispensable. In this letter, we present a full-body dynamics model for motor-actuated robotic fish, rigorously derived from Hamilton's principle. The model captures the continuously distributed elasticity of a deformable fish body undergoing large deformations and incorporates fluid-structure coupling effects, enabling self-propelled motion without prescribing kinematics. Preliminary open-loop simulations examine how actuation frequency and body stiffness influence the swimming speed and energy efficiency of the robotic fish. Closed-loop simulations further assess how stiffness distribution impacts the controller's velocity-tracking performance and energy efficiency. The results demonstrate the model's potential for performance evaluation and control optimization of soft robotic swimmers when stiffness is treated as a design variable.