In this paper, we design a resource management scheme to support stateful applications, which will be prevalent in 6G networks. Different from stateless applications, stateful applications require context data while executing computing tasks from user terminals (UTs). Using a multi-tier computing paradigm with servers deployed at the core network, gateways, and base stations to support stateful applications, we aim to optimize long-term resource reservation by jointly minimizing the usage of computing, storage, and communication resources and the cost from reconfiguring resource reservation. The coupling among different resources and the impact of UT mobility create challenges in resource management. To address the challenges, we develop digital twin (DT) empowered network planning with two elements, i.e., multi-resource reservation and resource reservation reconfiguration. First, DTs are designed for collecting UT status data, based on which UTs are grouped according to their mobility patterns. Second, an algorithm is proposed to customize resource reservation for different groups to satisfy their different resource demands. Last, a Meta-learning-based approach is developed to reconfigure resource reservation for balancing the network resource usage and the reconfiguration cost. Simulation results demonstrate that the proposed DT-empowered network planning outperforms benchmark frameworks by using less resources and incurring lower reconfiguration costs.