Unmanned aerial vehicles (UAV) are widely used for edge computing in poor infrastructure scenarios due to their deployment flexibility and mobility. In UAV-assisted edge computing systems, multiple UAVs can cooperate with the cloud to provide superior computing capability for diverse innovative services. However, many service-related computational tasks may fail due to the unreliability of UAVs and wireless transmission channels. Diverse solutions were proposed, but most of them employ timedriven strategies which introduce unwanted decision waiting delays. To address this problem, this paper focuses on a taskdriven reliability-aware cooperative offloading problem in UAVassisted edge-enhanced networks. The issue is formulated as an optimization problem which jointly optimizes UAV trajectories, offloading decisions, and transmission power, aiming to maximize the long-term average task success rate. Considering the discretecontinuous hybrid action space of the problem, a dependenceaware latent-space representation algorithm is proposed to represent discrete-continuous hybrid actions. Furthermore, we design a novel deep reinforcement learning scheme by combining the representation algorithm and a twin delayed deep deterministic policy gradient algorithm. We compared our proposed algorithm with four alternative solutions via simulations and a realistic Kubernetes testbed-based setup. The test results show how our scheme outperforms the other methods, ensuring significant improvements in terms of task success rate