The renaissance in aerospace technology positions satellite computing as a promising frontier. Due to the wide coverage of satellites, users in remote areas can request computing services even if there is no terrestrial infrastructure. Most existing studies focus on computation offloading, neglecting the service migration issues that state dependency induces. This paper focuses on user-dependent state information services. Due to the periodic motion of satellites, they will move away from the users. Users may frequently change satellites to achieve a lower delay, but this behavior can cause tremendous network pressure. Addressing this problem poses challenges due to the limited onboard resources and the high computational complexity of optimal node decision-making. Our approach first aims to minimize delay under the constraint of long-term costs. Second, we employ Lyapunov optimization to stabilize long-term costs and decouple the problem into a more manageable single-slot problem. Since the decomposition problem is NP-hard, we use a distributed approximation-based best response update to reduce computational complexity. Additionally, we propose a dynamic programming-based offline service migration algorithm, assuming complete information availability. The simulation results demonstrate the effectiveness of the proposed offline and online algorithms.