Smartwatch health sensor data is increasingly utilized in smart health applications and patient monitoring, including stress detection. However, such medical data often comprises sensitive personal information and is resource-intensive to acquire for research purposes. In response to this challenge, we introduce the privacy-aware synthetization of multi-sensor smartwatch health readings related to moments of stress. Our method involves the generation of synthetic sequence data through Generative Adversarial Networks (GANs), coupled with the implementation of Differential Privacy (DP) safeguards for protecting patient information during model training. To ensure the integrity of our synthetic data, we employ a range of quality assessments and monitor the plausibility between synthetic and original data. To test the usefulness, we create private machine learning models on a commonly used, albeit small, stress detection dataset, exploring strategies for enhancing the existing data foundation with our synthetic data. Through our GAN-based augmentation methods, we observe improvements in model performance, both in non-private (0.45% F1) and private (11.90-15.48% F1) training scenarios. We underline the potential of differentially private synthetic data in optimizing utility-privacy trade-offs, especially with limited availability of real training samples.