As the most important part of a smart city and long-standing challenging issue, a highly efficient smart logistic system has attracted a great deal of attention in recent years. In particular, unmanned aerial vehicles (UAVs) are ideal solutions for last-mile delivery scenarios in recent years due to their fast speed and easy deployment. However, because of the highly automatic delivery process, UAVs are still constrained by the limited payload, battery, and computing capacity for complex computational tasks. With the aid of the mobile edge computing (MEC) technology, UAVs can offload computational tasks to the MEC computational resources in various types of IoT environments. In spite of the task offloading which can enhance their task process capability, it also brings extra overhead, such as data transfer time and energy consumption. These extra overheads may significantly impact the efficiency and payload of UAV-based delivery systems. Therefore, taking the UAV last-mile delivery system with MEC as an example, this article investigates the energy-aware multi-UAV task computation management problem according to a realistic autonomous delivery network (ADNET). Specifically, we propose a computation management strategy, namely, the MEC-based task offloading and scheduling strategy (TOSS), to provide an integral approach covering both the static task offloading and scheduling algorithm, as well as the dynamic resource conflict resolution algorithm. Grounded on real-world scenarios, our experimental results show that TOSS can achieve a higher payload for UAVs by using minimum energy consumption and task makespan within the given constraints of the deadline compared to the state-of-the-art methods.