The satellite-terrestrial networks (STNs) enable global data delivery with low latency and high transmission rate utilizing the Low-Earth-Orbit (LEO) satellite constellation, especially for users in remote regions. However, the dynamic nature of STN presents two challenges to the quality of experience (QoE): 1) the dynamic topology in different time slots; 2) the dynamic path planning for each user. These raise unique problems for the optimization of data delivery paths in STN with service continuity guarantee. In this paper, we first model the STN by a time-varying graph to describe the data delivery in a certain service duration. An indicator called user satisfaction is constructed to measure delivery performance. Aiming at maximizing total user satisfaction, we reformulate the STN path planning problem by utilizing the spatial-temporal decomposition. Considering the path planning in each time slot, we modify the heuristic algorithm to optimize the allocation of transmission resources for different users. In the temporal dimension, we show that the path allocation results in previous time slots directly affect subsequent path calculation then propose an active time slot division algorithm to improve the STN service continuity. Excellent STN transmission performance is matched to the best fit time slot duration to obtain efficient and continuous data delivery. Experimental results show that the proposed STN data delivery optimization scheme achieves superior transmission performance with better service continuity.