This paper examines the motion of a nearly dynamically spherical rigid body rotating about its center of mass, where the body contains a cavity filled with a highly viscous fluid and includes a viscoelastic component. The system of motion equations is derived in standard form and further refined using a quadratic approximation with respect to a small parameter. The averaging method is then applied to the resulting nonlinear equations of motion. The Cauchy problem for the averaged system is analyzed, and the asymptotic approach enables the derivation of qualitative insights into the system’s behavior. The evolution of angular motion is described using both simplified averaged equations and numerical simulations. As a result, long-term solutions are obtained, characterizing the rigid body’s motion over an infinite time interval with asymptotically small error. Graphical representations of the solutions are provided and discussed. The work contributes to the study of spacecraft and satellite dynamics, including crew activity aboard vehicles. The findings are particularly relevant to the development of moving mass control systems and the analysis of spinning projectile motion.