This paper presents a novel routing algorithm designed for direct-to-cell satellite constellations in satellite networks. The algorithm aims to ensure stable Quality of Service (QoS) for users despite the highly dynamic nature of satellite links and network conditions. It comprises two key components: a terrestrial-satellite link selection model and an inter-satellite multi-hop routing decision model. To enhance decision-making, the algorithm incorporates a mechanism that interacts with large-scale digital twin constellations, offering agents with high-precision state data inputs and feedback for training and application support. The terrestrial-satellite link selection process is modeled as a constraint satisfaction problem, maximizing the Link Quality Indicator (LQI). Meanwhile, inter-satellite data forwarding decisions are determined utilizing an enhanced dynamic reward mechanism and a re-designed neural network in the DDQN model. Simulation results demonstrate the superiority of the Digital Twin-Assisted Double Deep Q-Network Routing (DT-DDQNR) algorithm over traditional routing algorithms in terms of overall network QoS performance, offering a promising solution for optimizing data transmission in integrated terrestrial-satellite networks.