This work introduces a numerical framework for addressing Fluid-Structure Interaction problems involving thin structures subject to finite strain deformations. The proposed approach utilizes an embedded mesh method to establish a coupling interface between the fluid and structural domains. The novelty of the work is the incorporation of a recently developed locking-free stabilized formulation of solid-shell elements to handle the structural domain. The framework employs established techniques to handle pressure jumps in the fluid domain across the embedding interface and enforce boundary conditions, such as discontinuous shape functions for the pressure unknowns designed to segregate nodal contributions of the cut elements, and Nitsche's method for the weak imposition of transmission conditions in the fluid. The present approach is validated through a series of benchmark cases in both 2D and 3D environments, progressively increasing in complexity. The results demonstrate good agreement with existing literature, establishing the presented framework as a viable method for addressing Fluid-Structure Interaction problems involving thin structures subject to large strains.