Being an up-and-coming application scenario of mobile edge computing (MEC), the post-disaster rescue suffers multitudinous computing-intensive tasks but unstably guaranteed network connectivity. In rescue environments, quality of service (QoS), such as task execution delay, energy consumption and battery state of health (SoH), is of significant meaning. This paper studies a multi-user post-disaster MEC environment with unstable 5G communication, where device-to-device (D2D) link communication and dynamic voltage and frequency scaling (DVFS) are adopted to balance each user's requirement for task delay and energy consumption. A battery degradation evaluation approach to prolong battery lifetime is also presented. The distributed optimization problem is formulated into a mixed cooperative-competitive (MCC) multi-agent Markov decision process (MAMDP) and is tackled with recurrent multi-agent Proximal Policy Optimization (rMAPPO). Extensive simulations and comprehensive comparisons with other representative algorithms clearly demonstrate the effectiveness of the proposed rMAPPO-based offloading scheme.