A practical dialogue system requires the capacity for ongoing skill acquisition and adaptability to new tasks while preserving prior knowledge. However, current methods for Continual Dialogue State Tracking (DST), a crucial function of dialogue systems, struggle with the catastrophic forgetting issue and knowledge transfer between tasks. We present TaSL, a novel framework for task skill localization and consolidation that enables effective knowledge transfer without relying on memory replay. TaSL uses a novel group-wise technique to pinpoint task-specific and task-shared areas. Additionally, a fine-grained skill consolidation strategy protects task-specific knowledge from being forgotten while updating shared knowledge for bi-directional knowledge transfer. As a result, TaSL strikes a balance between preserving previous knowledge and excelling at new tasks. Comprehensive experiments on various backbones highlight the significant performance improvements of TaSL, with a 7.6% absolute increase in Avg. JGA and an 11% absolute rise in BWT metrics over existing state-of-the-art methods. The source code is provided for reproducibility.