Space–air–aqua integrated network (SAAIN) is an emerging maritime network architecture to support reliable and timely communications. Considering the computation capability and information security in maritime transportation systems, this work proposes a double-edge secure offloading scheme, where both base-station (BS) and satellites provide secure mobile-edge computing services for delay-sensitive applications. Specifically, maritime mobile users may offload their computation tasks adaptively to the BS or satellites securely relayed by unmanned aerial vehicles (UAVs). To minimize offloading delay, we formulate an optimization problem to allocate the transmit power cooperatively via UAVs’ trajectory optimization. Moreover, jamming UAVs are deployed to protect the offloading process. For such a nonconvex optimization problem, two iterative algorithms are proposed to determine the transmit power and design the UAVs’ trajectories. Numerical results show the effectiveness of the proposed scheme in terms of offloading delay.