The traditional Dialogue State Tracking (DST) problem aims to track user preferences and intents in user-agent conversations. While sufficient for task-oriented dialogue systems supporting narrow domain applications, the advent of Large Language Model (LLM)-based chat systems has introduced many real-world intricacies in open-domain dialogues. These intricacies manifest in the form of increased complexity in contextual interactions, extended dialogue sessions encompassing a diverse array of topics, and more frequent contextual shifts. To handle these intricacies arising from evolving LLM-based chat systems, we propose joint dialogue segmentation and state tracking per segment in open-domain dialogue systems. Assuming a zero-shot setting appropriate to a true open-domain dialogue system, we propose S3-DST, a structured prompting technique that harnesses Pre-Analytical Recollection, a novel grounding mechanism we designed for improving long context tracking. To demonstrate the efficacy of our proposed approach in joint segmentation and state tracking, we evaluate S3-DST on a proprietary anonymized open-domain dialogue dataset, as well as publicly available DST and segmentation datasets. Across all datasets and settings, S3-DST consistently outperforms the state-of-the-art, demonstrating its potency and robustness the next generation of LLM-based chat systems.