Unsupervised Domain Adaptation (UDA) is essential for deploying medical segmentation models across diverse clinical environments. Existing methods are fundamentally limited, suffering from semantically unaware feature alignment that results in poor distributional fidelity and from pseudo-label validation that disregards global anatomical constraints, thus failing to prevent the formation of globally implausible structures. To address these issues, we propose SHAPE (Structure-aware Hierarchical Unsupervised Domain Adaptation with Plausibility Evaluation), a framework that reframes adaptation towards global anatomical plausibility. Built on a DINOv3 foundation, its Hierarchical Feature Modulation (HFM) module first generates features with both high fidelity and class-awareness. This shifts the core challenge to robustly validating pseudo-labels. To augment conventional pixel-level validation, we introduce Hypergraph Plausibility Estimation (HPE), which leverages hypergraphs to assess the global anatomical plausibility that standard graphs cannot capture. This is complemented by Structural Anomaly Pruning (SAP) to purge remaining artifacts via cross-view stability. SHAPE significantly outperforms prior methods on cardiac and abdominal cross-modality benchmarks, achieving state-of-the-art average Dice scores of 90.08% (MRI->CT) and 78.51% (CT->MRI) on cardiac data, and 87.48% (MRI->CT) and 86.89% (CT->MRI) on abdominal data. The code is available at https://github.com/BioMedIA-repo/SHAPE.