With the swift progress of Internet of Things (IoT) technologies, the number of IoT devices has grown exponentially, leading to an increasing demand for computational power and system stability. Mobile edge computing (MEC) is a powerful solution that allows IoT devices to offload data to the edge for computing. In situations involving disasters or complex terrains, establishing ground-based stations may be challenging in providing computational services. Edge computing frameworks built with unmanned aerial vehicles (UAVs) and high-altitude platforms (HAPs) can provide airborne computational services for IoT devices situated in environments with disasters or complex terrains. In this article, we design a three-tier framework consisting of ground users (GUs), UAVs, and HAP, offering MEC services for GUs. Considering the randomness and dynamism of task arrivals and the wireless communication quality of devices, we propose an algorithm supporting nonorthogonal multiple access (NOMA) communication in aerial access networks. The objective of the algorithm is to reduce the overall energy consumption of the system while ensuring system stability. Employing stochastic optimization techniques, we convert the task offloading and resource allocation problem into several parallel solvable subproblems. We also provide a theoretical analysis of the algorithm. Through a series of comparative experiments, we demonstrate the feasibility and effectiveness of our proposed dynamic energy-efficient computation offloading (DEECO) algorithm.