Sentiment analysis plays a crucial role in various domains, such as business intelligence and financial forecasting. Large language models (LLMs) have become a popular paradigm for sentiment analysis, leveraging multi-task learning to address specific tasks concurrently. However, LLMs with fine-tuning for sentiment analysis often underperforms due to the inherent challenges in managing diverse task complexities. Moreover, constant-weight approaches in multi-task learning struggle to adapt to variations in data characteristics, further complicating model effectiveness. To address these issues, we propose a novel multi-task learning framework with a dynamic adaptive optimization (DAO) module. This module is designed as a plug-and-play component that can be seamlessly integrated into existing models, providing an effective and flexible solution for multi-task learning. The key component of the DAO module is dynamic adaptive loss, which dynamically adjusts the weights assigned to different tasks based on their relative importance and data characteristics during training. Sentiment analyses on a standard and customized financial text dataset demonstrate that the proposed framework achieves superior performance. Specifically, this work improves the Mean Squared Error (MSE) and Accuracy (ACC) by 15.58% and 1.24% respectively, compared with previous work.