In this paper, we investigate the joint design of beamforming vectors and trajectory for unmanned aerial vehicles (UAVs) in integrated sensing and communications networks, aiming to maximize the achievable covert rate (ACR) for legitimate users against multiple passive wardens. Considering the worst-case scenario, where the wardens strategically select optimal decision thresholds, we derive the minimum detection error probability and incorporate covertness constraints within the beamforming scheme. Our approach entails formulating the design as a non-convex optimization problem for maximizing the average ACR along the UAV trajectory. The formulation takes into account various practical constraints, such as the maximum transmit power, UAV flight speed limitations, minimum beamforming gain towards sensing targets, and the detection probability threshold for wardens. To address this intricate problem, we propose a block coordinate descent-based optimization algorithm. This algorithm alternates between updating beamforming vectors and UAV trajectories, offering a high-quality suboptimal solution to the original problem. Theoretical analyses reveal that when the detection probability threshold is sufficiently small, a linear correlation emerges between the maximum relative variation ratio in the average received signal power at wardens under two hypotheses and the detection probability. Furthermore, to enhance covertness against the wardens, it is necessary to either decrease the projection of information beamforming covariance matrix or increase the projection of sensing beamforming covariance matrix onto the subspace spanned by the eavesdropping channel vectors. Finally, extensive simulations are presented to validate the covert performance enhancements of our proposed methodology, compared with various baseline schemes adopting existing approaches.