This paper focuses on the development of the partitioned solution method (PSM) for analyzing the stability behavior of doubly-curved shallow orthotropic panels under external pressure, covering both the buckling and postbuckling responses. Adjacent equilibrium method (AEM) is used to verify the developed PSM method and the associated stability results. The equilibrium and compatibility equations are derived using Donnell-type thin shell theory, with the Airy stress function and the out-of-plane displacement as unknowns. Based on AEM and PSM, both an eigenvalue problem and non-linear algebraic equations are obtained which are used as the basis for the stability criteria, respectively. Results obtained from those two methods are presented and compared with each other for a few arbitrary sets of system parameters, wherein no postbuckling solutions are presented with AEM. The influence of the boundary conditions on the stability behavior is also investigated using the PSM.