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This paper leverages large-language models (LLMs) to experimentally determine strategies for scaling up social media annotation and stance detection of health information, with HPV vaccine-related tweets as a case study. We examine both conventional fine-tuning and emergent in-context learning methods, systematically varying strategies of prompt engineering and in-context learning across widely used LLMs and their variants (e.g., GPT-4, Mistral, Llama 3, and Flan- UL2). Specifically, we varied prompt template design, shot sampling methods, and shot quantity to detect stance on HPV vaccination. Our findings reveal that (a) in-context learning outperformed fine-tuning in stance detection for HPV vaccine social media content; (b) increasing shot quantity does not necessarily enhance performance across models; (c) stratified sampling often outperforms random sampling, with the performance gap more pronounced in smaller model variants, and (d) LLMs and their variants present differing sensitivity to in-context learning conditions. This study highlights the potential and provides an applicable approach for applying LLMs to research on social media annotation and stance detection of health information.