The fatigue loads experienced by components and the motion of the platform in floating wind turbines are likely influenced by their operation within a limited power state; however, the corresponding effects remain inadequately understood. This study investigates a 5 MW semi-submersible floating wind turbine, aiming to provide a comprehensive analysis of fatigue loads on critical components and platform motion under various power regulation modes. To achieve this objective, we employed the NREL 5 MW semi-submersible wind turbine model, generated three-dimensional wind fields using TurbSim, and conducted dynamic simulations with OpenFAST, utilizing MLife software for post-processing of fatigue load analysis. We assessed the fatigue loads of four key components—the blade root, tower base, drivetrain, and mooring cables—alongside the platform motion under varying generator speeds, yaw angles, and active power levels. Additionally, correlation analysis was performed to explore the relationship between component fatigue loads and platform motion under different yaw conditions. The results demonstrate that both generator speed and yaw angle significantly influence fatigue loads and platform motion, while the effect of active power appears to be relatively minor. Specifically, an increase in generator speed markedly elevates fatigue loads on the blade root and drivetrain, while concurrently reducing loads on the tower base and certain mooring lines. The analysis further reveals that, under various yaw angles, the fatigue loads on symmetrically positioned mooring cables exhibit a symmetric response, impacting fatigue damage in the blade root and tower base moments. Moreover, the relationship between maximum horizontal platform displacement and fatigue loads varies. The findings of this study provide critical insights into the operational optimization of floating wind turbines.