The growth in artificial intelligence (AI) technology has attracted substantial interests in latency-aware task offloading of mobile edge computing (MEC)-namely, minimizing service latency. Additionally, the use of MEC systems poses an additional problem arising from limited battery resources of MDs. This paper tackles the pressing challenge of latency-aware distributed task offloading optimization, where user association (UA), resource allocation (RA), full-task offloading, and battery of mobile devices (MDs) are jointly considered. In existing studies, joint optimization of overall task offloading and UA is seldom considered due to the complexity of combinatorial optimization problems, and in cases where it is considered, linear objective functions such as power consumption are adopted. Revolutionizing the realm of MEC, our objective includes all major components contributing to users' quality of experience, including latency and energy consumption. To achieve this, we first formulate an NP-hard combinatorial problem, where the objective function comprises three elements: communication latency, computation latency, and battery usage. We derive a closed-form RA solution of the problem; next, we provide a distributed pricing-based UA solution. We simulate the proposed algorithm for various resource-intensive tasks. Our numerical results show that the proposed method Pareto-dominates baseline methods. More specifically, the results demonstrate that the proposed method can outperform baseline methods by 1.62 times shorter latency with 41.2% less energy consumption.