Hardware bottlenecks can throttle smart device (SD) performance when executing computation-intensive and delay-sensitive applications. Hence, task offloading can be used to transfer computation-intensive tasks to an external server or processor in Mobile Edge Computing. However, in this approach, the offloaded task can be useless when a process is significantly delayed or a deadline has expired. Due to the uncertain task processing via offloading, it is challenging for each SD to determine its offloading decision (whether to local or remote and drop). This study proposes a deep-reinforcement-learning-based offloading scheduler (DRL-OS) that considers the energy balance in selecting the method for performing a task, such as local computing, offloading, or dropping. The proposed DRL-OS is based on the double dueling deep Q-network (D3QN) and selects an appropriate action by learning the task size, deadline, queue, and residual battery charge. The average battery level, drop rate, and average latency of the DRL-OS were measured in simulations to analyze the scheduler performance. The DRL-OS exhibits a lower average battery level (up to 54%) and lower drop rate (up to 42.5%) than existing schemes. The scheduler also achieves a lower average latency of 0.01 to >0.25 s, despite subtle case-wise differences in the average latency.