Large volumes of medical data remain underutilized because centralizing distributed data is often infeasible due to strict privacy regulations and institutional constraints. In addition, models trained in centralized settings frequently fail to generalize across clinical sites because of heterogeneity in imaging protocols and continuously evolving data distributions arising from differences in scanners, acquisition parameters, and patient populations. Federated learning offers a promising solution by enabling collaborative model training without sharing raw data. However, incorporating differential privacy into federated learning, while essential for privacy guarantees, often leads to degraded accuracy, unstable convergence, and reduced generalization. In this work, we propose an adaptive differentially private federated learning (ADP-FL) framework for medical image segmentation that dynamically adjusts privacy mechanisms to better balance the privacy-utility trade-off. The proposed approach stabilizes training, significantly improves Dice scores and segmentation boundary quality, and maintains rigorous privacy guarantees. We evaluated ADP-FL across diverse imaging modalities and segmentation tasks, including skin lesion segmentation in dermoscopic images, kidney tumor segmentation in 3D CT scans, and brain tumor segmentation in multi-parametric MRI. Compared with conventional federated learning and standard differentially private federated learning, ADP-FL consistently achieves higher accuracy, improved boundary delineation, faster convergence, and greater training stability, with performance approaching that of non-private federated learning under the same privacy budgets. These results demonstrate the practical viability of ADP-FL for high-performance, privacy-preserving medical image segmentation in real-world federated settings.