<p>The paper examines the motion of a nearly dynamically spherical rigid body rotating about its center of mass, where the body subjected to constant body-fixed torques and contains a cavity filled with a highly viscous fluid and includes a viscoelastic element. 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 applied to the nonlinear equations of motion. The Cauchy problem for the system determined after averaging is investigated. The evolution of angular motion is described using averaged equations and numerical simulation. The advantage of this paper is that it offers solutions that describe the evolution of a rigid body motion over infinite time interval with an 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 and is particularly relevant to the development of moving mass control.</p>