Floating offshore wind turbines (FOWTs) are subject to undesirable platform motion and a significant increases in fatigue loads compared to their onshore counterparts. We have recently proposed using the fishing line artificial muscle (FLAM) actuators to realize active mooring line force control (AMLFC) for platform stabilization and thus load reduction, which features a compact design and no need for turbine redesign. However, as for the thermally activated FLAM actuators, a major control challenge lies in the asymmetric dynamics for the heating and the cooling half cycle of operation. In this paper, for a tension-leg platform (TLP) based FOWT with FLAM actuator based AMLFC, a hybrid dynamic model is obtained with platform pitch and roll degrees-of-freedom included. Then a hybrid model predictive control (HMPC) strategy is proposed for platform motion stabilization, with preview information on incoming wind and wave. A move blocking scheme is used to achieve reasonable computational efficiency. Fatigue, aerodynamics, structures, and turbulence (FAST) based simulation study is performed using the National Renewable Energy Laboratory (NREL) 5 MW wind turbine model. Under different combinations of wind speed, wave height and wind directions, simulation results show that the proposed control strategy can significantly reduce the platform roll and tower-base side-to-side bending moment, with a mild level of actuator power consumption.