A two-dimensional fully nonlinear potential flow model is employed to investigate nonlinear stochastic responses of an experimental fluid-structure interaction system that includes both single-degree-of-freedom surge-only and two-degree-of-freedom surge-heave coupled motions. Sources of nonlinearity include free surface boundary, fluid-structure interaction, and large geometry in the structural restoring force. Random waves performed in the tests include nearly periodic, periodic with band-limited noise, and narrow band. The structural responses observed can be categorized as nearly deterministic (harmonic, sub- and super-harmonic), noisy periodic, and random. Transition phenomena between coexisting response attractors are also identified. An implicit boundary condition upholding the instantaneous equilibrium between the fluid and structure using a mixed Eulerian-Lagrangian method is employed. Numerical model predictions are calibrated and validated via the experimental results under the three types of wave conditions. Extensive simulations are conducted to identify the response characteristics and the effects of random perturbations on nonlinear responses near primary and secondary resonances.