Dynamic scheduling for flexible job shops under machine breakdown is a complex and challenging problem due to its valuable application in real-life productions. However, prior studies have struggled to perform well in changeable scenarios. To address this challenge, this paper introduces a dual-objective deep reinforcement learning (DRL) to solve this problem. This algorithm is based on the Double Deep Q-network (DDQN) and incorporates the attention mechanism. It decouples action relationships in the action space to reduce problem dimensionality and introduces an adaptive weighting method in agent decision-making to obtain high-quality Pareto front solutions. The algorithm is evaluated on a set of benchmark instances and compared with state-of-the-art algorithms. The experimental results show that the proposed algorithm outperforms the state-of-the-art algorithms regarding machine offset and total tardiness, demonstrating more excellent stability and higher-quality solutions. At the same time, the actual use of the algorithm is verified using cases from real enterprises, and the results are still better than those of the multi-objective meta-heuristic algorithm.