Analyzing different white blood cell (WBC) classes is essential for human health monitoring, making accurate segmentation and classification crucial for diagnosing blood-related conditions. Existing WBC segmentation systems mainly rely on convolution neural networks (CNNs) and Transformers. Unfortunately, they are unable to simultaneously capture global context and local information. Similarly, existing WBC classification systems fail to appropriately focus on the relevant regions of WBC images. Additionally, the processes of WBC classification and segmentation are intertwined, but they are not properly synergized in the literature. These three issues have limited the efficacy of existing WBC segmentation and classification systems. Our proposed system, SANGAM, improves the efficacy of WBC segmentation and classification by addressing these issues. Specifically, it integrates the local learning capabilities of CNNs with the global context learning capabilities of Transformers to enhance WBC segmentation. It also improves WBC classification by providing more attention to the relevant areas. Furthermore, it synergizes WBC segmentation and classification. Our experimental results, conducted on publicly available datasets, reveal that SANGAM outperforms existing well-known WBC segmentation and classification systems. Additionally, it advocates for the appropriate integration of CNNs and Transformers in WBC segmentation, providing attention to relevant regions in WBC classification, and synergizing WBC classification and segmentation.