Respiratory infectious diseases continuously threaten global public health, and timely accurate prediction of epidemiological indicators is essential for public health intervention. Traditional forecasting models are limited in fusing multimodal data and dynamic contextual information, while existing LLM-based methods mostly depend on static prompts and fail to cope with low-quality textual data. This paper systematically explores context-aware time-series forecasting frameworks for respiratory infectious diseases, integrating structured epidemiological metrics with unstructured textual context. Domain-adapted parameter-efficient fine-tuning is employed to adapt an 8B-parameter LLM under a single-GPU local deployment setting, enabling seamless fusion of temporal and textual information. A task-oriented text quality assessment and filtering module is introduced to examine whether low-relevance contextual texts affect LLM-based epidemic forecasting and to improve the signal-to-noise ratio of public health textual context. Experiments on two authoritative datasets demonstrate strong performance of the optimized framework. The fine-tuned Llama3-8B model achieves the best performance on the PandemicLLM dataset, while showing competitive performance on the unfiltered Time-MMD dataset, where the strongest temporal model remains superior. Ablation studies demonstrate the complementary effect of static and dynamic contextual features, and verify that text filtering can effectively suppress noise and boost model performance. These findings demonstrate the potential of context-aware LLM forecasting under the evaluated U.S. public health data settings and highlight the importance of contextual data quality for multimodal epidemic prediction.