Satellite networks with edge computing servers promise to provide ubiquitous and low-latency computing services for the Internet of Things (IoT) applications in the future satellite-terrestrial integrated network (STIN). For some emerging IoT applications, the services require real-time user-dependent state information, such as time-varying task states and user-specific configurations, to maintain service continuity. Service migration is crucial for dynamic task offloading to synchronize the user-dependent state information between computing servers. However, how to offload computing tasks at low latency with the impact of service migration remains challenging due to the high-speed movement and load imbalance of low Earth orbit (LEO) satellite networks. In this work, we investigate the task offloading problem with service migration for satellite edge computing (SEC) using inter-satellite cooperation. Facing dynamic service requirements with limited on-board bandwidth, energy, and storage resources of satellite networks, we formulate the problem with the aim of minimizing the service delay to optimize the offloading path selection. By leveraging a deep reinforcement learning (DRL) approach, we propose a distributed scheme based on the Dueling-Double-Deep-Q-Learning (D3QN) algorithm. Simulation results show that the proposed scheme can effectively reduce the service delay, and outperform the benchmark algorithms.INDEX TERMS Satellite edge computing (SEC), task offloading, service migration, deep reinforcement learning (DRL).