This paper introduces a network architecture that integrates the space-air-ground integrated network with mobile edge computing (MEC) and orbital edge computing to advance sixth-generation communication systems. The proposed system employs unmanned aerial vehicles equipped with reconfigurable intelligent surfaces and satellite-based MEC to optimize resource management in complex, dynamic environments. By efficiently managing resources such as bandwidth and computational power at both base stations and low Earth orbit satellites, while making offloading decisions, the system aims to minimize utility costs while meeting stringent performance requirements. We utilize a long short-term memory (LSTM)-enhanced deep deterministic policy gradient (DDPG) algorithm to solve the formulated nonlinear programming problem, enabling dynamic and adaptive resource management. The LSTM-enhanced DDPG improves convergence speed by 44.44% compared to conventional DDPG, significantly enhancing cost efficiency. Simulation results validate the robustness of the proposed method against state-of-the-art approaches.