Handwritten signature verification is a crucial aspect of identity authentication, with applications in various domains such as finance and e-commerce. However, achieving high accuracy in signature verification remains challenging due to intra-user variability and the risk of forgery. This paper introduces a novel approach for dynamic signature verification: the Temporal-Spatial Graph Attention Transformer (TS-GATR). TS-GATR combines the Graph Attention Network (GAT) and the Gated Recurrent Unit (GRU) to model both spatial and temporal dependencies in signature data. TS-GATR enhances verification performance by representing signatures as graphs, where each node captures dynamic features (e.g. position, velocity, pressure), and by using attention mechanisms to model their complex relationships. The proposed method further employs a Dual-Graph Attention Transformer (DGATR) module, which utilizes k-step and k-nearest neighbor adjacency graphs to model local and global spatial features, respectively. To capture long-term temporal dependencies, the model integrates GRU, thereby enhancing its ability to learn dynamic features during signature verification. Comprehensive experiments conducted on benchmark datasets such as MSDS and DeepSignDB show that TS-GATR surpasses current state-of-the-art approaches, consistently achieving lower Equal Error Rates (EER) across various scenarios.