Prior research on fine-grained complaint analysis has largely focused on short, context-limited inputs such as tweets or product reviews. In this work, we advance the field by introducing a paradigm that leverages multi-turn customer-support dialogues as a richer medium for capturing nuanced dissatisfaction. Such conversations provide dynamic signals, emotional shifts, iterative follow-ups, and detailed issue descriptions that enable more accurate detection of aspect categories (e.g., service quality, software issues) and severity levels (e.g., disapproval, accusation). To operationalize this problem formulation, we introduce CompSense, a multi-task Mixture-of-Experts (MoE) framework enriched with commonsense-aware contextualization. The framework further incorporates severity-aware gating to adapt expert selection based on complaint intensity and supervised contrastive learning to structure representations by clustering similar aspect–severity patterns while separating ambiguous cases. Extensive evaluations show that CompSense consistently outperforms both task-specific conversational models and general purpose LLM baselines, underscoring the effectiveness of the proposed framework. This work marks a step toward practical, real world systems capable of sophisticated conversational complaint analysis (Resources are available at: https://github.com/sarmistha-D/CompSense .).