Reinforcement learning (RL) in text-based games has developed rapidly and achieved promising results. However, little effort has been expended to design agents that pursue objectives while behaving morally, which is a critical issue in the field of autonomous agents. In this paper, we propose a general framework named Moral Awareness Adaptive Learning (MorAL) that enhances the morality capacity of an agent using a plugin moral-aware learning model. The framework allows the agent to execute task learning and morality learning adaptively. The agent selects trajectories from past experiences during task learning. Meanwhile, the trajectories are used to conduct self-imitation learning with a moral-enhanced objective. In order to achieve the trade-off between morality and task progress, the agent uses the combination of task policy and moral policy for action selection. We evaluate on the Jiminy Cricket benchmark, a set of text-based games with various scenes and dense morality annotations. Our experiments demonstrate that, compared with strong contemporary value alignment approaches, the proposed framework improves task performance while reducing immoral behaviours in various games.