The widespread dissemination of rumors on social media has a significant impact on people's lives, potentially leading to public panic and fear. Rumors often evoke specific sentiments, resonating with readers and prompting sharing. To effectively detect and track rumors, it is essential to observe the fine-grained sentiments of both source and response message pairs as the rumor evolves over time. However, current rumor detection methods fail to account for this aspect. In this paper, we propose MSuf, the first multi-task suffix learning framework for rumor detection and tracking using time series dual (coupled) sentiments. MSuf includes three modules: (1) an LLM to extract sentiment intensity features and sort them chronologically; (2) a module that fuses the sorted sentiment features with their source text word embeddings to obtain an aligned embedding; (3) two hard prompts are combined with the aligned vector to perform rumor detection and sentiment analysis using one frozen LLM. MSuf effectively enhances the performance of LLMs for rumor detection with only minimal parameter fine-tuning. Evaluating MSuf on four rumor detection benchmarks, we find significant improvements compared to other emotion-based methods.