Understanding how climate change affects us and learning about available solutions are key steps toward empowering individuals and communities to mitigate and adapt to it. As Large Language Models (LLMs) rise in popularity, it is necessary to assess their capability in this domain. In this study, we present a comprehensive evaluation framework, grounded in science communication principles, to analyze LLM responses to climate change topics. Our framework emphasizes both the presentational and epistemological adequacy of answers, offering a fine-grained analysis of LLM generations. Spanning 8 dimensions, our framework discerns up to 30 distinct issues in model outputs. The task is a real-world example of a growing number of challenging problems where AI can complement and lift human performance. We introduce a novel and practical protocol for scalable oversight that uses AI Assistance and relies on raters with relevant educational backgrounds. We evaluate several recent LLMs and conduct a comprehensive analysis of the results, shedding light on both the potential and the limitations of LLMs in the realm of climate communication.