In this paper, a new sandwich beam element is introduced for analyzing sandwich beam structures with a flexible core and partially delaminated regions. In this element, interfaces between the core and face sheets are modeled by two independent layers. The model uses a high-order sandwich panel theory to consider flexibility of the core with nonlinearities associated with geometry and real contact characteristics of the delaminated regions. The proposed motion field takes advantage of both displacement and displacement gradients in the core boundaries. Therefore, the kinematics allow continuity conditions for displacements and rotations at the interfaces to be exactly satisfied in fully bounded and delamination regions. By using finite element (FE) formulation with Hermite shape functions, elemental vectors and matrices are derived in the framework of Hamilton's principle. FE governing equations are solved by Newton-Raphson iterative scheme. A 2D FE method is also developed to verify predictions of the sandwich model for various delamination cases. Comparison studies show that results from both sandwich and 2D FE models are in a good agreement. They can predict large-deformation results for the sandwich behaviors much better than the simplified model available in the literature. A set of parametric study is devoted to provide an insight into the influence of boundary condition, number and position of delaminated regions on deformations, stresses and instability of fully clamped and cantilever sandwich beams. The developed formulation is not only more computationally efficient than 2D models usually used for such analysis, but also at the same time is accurate, simple and robust. It is also found that modeling of the delaminated zone core and stress distribution at each interface independently is crucial to accurately analyze instability behaviors of sandwich structures.