In task-oriented dialogue systems, Dialogue State Tracking (DST) aims to extract users' intentions from the dialogue history. Currently, most existing approaches suffer from error propagation and are unable to dynamically select relevant information when utilizing previous dialogue states. Moreover, the relations between the updates of different slots provide vital clues for DST. However, the existing approaches rely only on predefined graphs to indirectly capture the relations. In this paper, we propose a Dialogue State Distillation Network (DSDN) to utilize relevant information of previous dialogue states and migrate the gap of utilization between training and testing. Thus, it can dynamically exploit previous dialogue states and avoid introducing error propagation simultaneously. Further, we propose an inter-slot contrastive learning loss to effectively capture the slot co-update relations from dialogue context. Experiments are conducted on the widely used MultiWOZ 2.0 and MultiWOZ 2.1 datasets. The experimental results show that our proposed model achieves the state-of-the-art performance for DST.