<p>In this research, a comprehensive investigation of ion acoustic waves within plasma environments, utilizing the Fermi-Dirac electron distribution, is presented. The reductive perturbation technique is used to derive the three-dimensional Davey-Stewartson equation, and analytical solutions are found in Jacobi elliptic cnoidal and solitary traveling wave forms. The dynamic behaviors of the solutions are then analyzed and studied parametrically. It is believed that current research can significantly contribute to present understanding of nonlinear wave behavior in multispecies warm and dense plasma systems and can have diverse applications to astrophysical and laboratory plasmas. Our investigation using a detailed numerical analysis focuses on the impact of the generalized equation of state (EoS) for Fermi-Dirac electron-ion plasma with three-dimensional Davey-Stewartson solutions with in-depth analysis of the modulation instability of the ion acoustic excitations. Current study incorporates variety of plasma parameters, such as the normalized chemical potential and electron temperature, also exploring electron number density effect on the wide range in electron density-temperature regime. This work advances current understanding of traveling wave behavior, the modulation stability corresponding to the propagation angle, and the instability growth rate providing a comprehensive insight into ion acoustic wave dynamics in plasmas.</p>