Federated learning (FL) is a distributed machine learning approach that allows multiple clients to collaboratively train a model without sharing their raw data. To prevent sensitive information from being inferred through the model updates shared in FL, differentially private federated learning (DPFL) has been proposed. DPFL ensures formal and rigorous privacy protection in FL by clipping and adding random noise to the shared model updates. However, the existing DPFL methods often result in severe model utility degradation, especially in settings with data heterogeneity. To enhance model utility, we propose a novel DPFL method named DP$^2$-FedSAM: Differentially Private and Personalized Federated Learning with Sharpness-Aware Minimization. DP$^2$-FedSAM leverages personalized partial model-sharing and sharpness-aware minimization optimizer to mitigate the adverse impact of noise addition and clipping, thereby significantly improving model utility without sacrificing privacy. From a theoretical perspective, we provide a rigorous theoretical analysis of the privacy and convergence guarantees of our proposed method. To evaluate the effectiveness of DP$^2$-FedSAM, we conduct extensive evaluations based on common benchmark datasets. Our results verify that our method improves the privacy-utility trade-off compared to the existing DPFL methods, particularly in heterogeneous data settings.