Modeling how ocean waves propagate and interact with offshore structures has been studied for decades, yet several important scientific challenges remain, particularly when using unfitted/embedded/immersed numerical techniques. Assuming the fluid is incompressible, inviscid, and irrotational, the wave dynamics follow fully nonlinear potential flow (FNPF), where each time step requires solving a mass-conserving Laplace problem coupled with two nonlinear free-surface evolution conditions. The computational domain is highly time-dependent due to the moving free surface and exhibits non-affine curvature. Combined with complex bathymetry and fully or partially submerged bodies (rigid, forced, or freely moving), mesh generation becomes both essential and time-consuming. To address this, we employ a high-order variant of the shifted boundary method (SBM) [1] with a polynomial correction [2], which is applied to the FNPF problem in [3]. The spatial discretization relies on high-order spectral elements (SEM) [4], which have been successfully employed in water-wave modeling for the past decade [5]. In this unfitted high-order scheme, the true geometry is embedded in a regular quadrilateral background mesh, from which an affine surrogate domain is constructed. Boundary conditions are then shifted to the surrogate boundaries while retaining optimal convergence. We present recent results using this FNPF-SEM-SBM formulation for wave-structure interaction problems, including waves over variable bathymetry, interactions with fully and partially submerged bodies, waves generated by forced motion, and related cases. Particular attention is given to the free surface and how its nonlinear conditions are affected by the unfitted Laplace domain.