An advanced model for analyzing sandwich plates and shells with anisotropic laminated composite face sheets and a weakly orthotropic core is presented, focusing on the determination of eigenfrequencies. The model is based on shallow shell theory and accounts for various geometric and material parameters. Key factors such as panel geometry, ply angle, stacking sequence of the face sheets, orthotropicity ratio of the core, face sheet thickness, and aspect ratio are all shown to influence the eigenfrequencies of the sandwich structure. Owing to the nature of the governing equations, a closed-form solution is achievable, yielding accurate results. The model’s validity is confirmed through comparison with established results from the literature, leading to meaningful conclusions.