This paper explores the nonlinear dynamic responses of a floating off-shore wind turbine (FOWT) subjected to the coupled effects of large-amplitude waves and an underlying uniform current. Using a Euler-Lagrangian analytical model, a 5MW OC3 spar-type floating wind turbine is coupled with a nonlinear hydrodynamic mooring model to evaluate structural responses and calculate cable fairlead forces. We demonstrate that the combination of current and large-amplitude waves significantly impacts the FOWT's responses and mooring forces. The coupling load generates larger vertical and horizontal displacements of the FOWT, as well as increased cable forces, compared to scenarios involving only waves or linear waves and current. Additionally, the direction of the current must be considered to ensure accurate results. The analysis encompasses various current direction, allowing for a comprehensive investigation of the relationship between wave conditions and current strength, which can be used to explore the internal mechanisms of the FOWT's responses from various wave conditions.