This paper considers covert communications in the context of unmanned aerial vehicle (UAV) networks where the UAV is employed as a transmitter to covertly disseminate data to a group of legitimate receivers on the ground, while ensuring that the data dissemination is not detected by the wardens. Considering the endurance time limit of UAV, our goal is to minimize the UAV's mission completion time by jointly optimizing the trajectory of UAV and the ground receivers' schedule. Since the environment considered is dynamic, the optimization problem is firstly modeled as a Markov decision process. Taking the advantage of the deep reinforcement learning (DRL) to learn dynamically from the environment, we propose a twin-delayed deep deterministic policy gradient (TD3) aided covert data dissemination (TD3-CDD) algorithm. In particular, we developed an advanced reward design mechanism to ensure the effectiveness of the constraints on UAV. Our examination shows that the TD3-CDD algorithm enables the UAV to complete covert data dissemination in a shorter time than a benchmark scheme.