Seams encode rich structural information about garments but are frequently partially observable in robotic manipulation scenarios. To robustly leverage seam information, we propose a Seam-to-Graph network based on graph neural networks and attention mechanisms. This network maps unstructured seam observations to a topology-encoded structural skeleton graph for real-time garment state estimation. Using this skeleton-graph-based state estimation, we design a deformation-aware, hierarchical visual servoing controller for garment configuration alignment. We implement this controller on a bimanual robot system to load a garment onto a screen printing platen and to align it to the desired configuration precisely. Real-robot experiments demonstrate that the robot using the proposed method not only achieves human-level alignment accuracy with reduced variance in alignment error but is also robust to different garments. These results demonstrate that the use of seam information is effective for garment manipulation.