The explosive growth in wireless service demand has prompted the evolution of integrated satellite-terrestrial networks (ISTNs) to overcome the limitations of traditional terrestrial networks (TNs) in terms of coverage, spectrum efficiency, and deployment cost. Particularly, leveraging LEO satellites and dynamic spectrum sharing (DSS), ISTNs offer promising solutions but face significant challenges due to diverse terrestrial environments, user and satellite mobility, and long propagation LEO-to-ground distance. To address these challenges, digitial-twin (DT) has emerged as a promising technology to offer virtual replicas of real-world systems, facilitating prediction for resource management. In this work, we study a time-window-based DT-aided DSS framework for ISTNs, enabling joint long-term and short-term resource decisions to reduce system congestion. Based on that, two optimization problems are formulated, which aim to optimize resource management using DT information and to refine obtained solutions with actual real-time information, respectively. To efficiently solve these problems, we proposed algorithms using compressed-sensing-based and successive convex approximation techniques. Simulation results using actual traffic data and the London 3D map demonstrate the superiority in terms of congestion minimization of our proposed algorithms compared to benchmarks. Additionally, it shows the adaptation ability and practical feasibility of our proposed solutions.