Cross-domain fake news detection (CD-FND) transfers knowledge from a source domain to a target domain and is crucial for real-world fake news mitigation. This task becomes particularly important yet more challenging when the target domain is previously unseen (e.g., the COVID-19 outbreak or the Russia-Ukraine war). However, existing CD-FND methods overlook such scenarios and consequently suffer from the following two key limitations: (1) insufficient modeling of high-level semantics in news and user engagements; and (2) scarcity of labeled data in unseen domains. Targeting these limitations, we find that large language models (LLMs) offer strong potential for CD-FND on unseen domains, yet their effective use remains non-trivial. Nevertheless, two key challenges arise: (1) how to capture high-level semantics from both news content and user engagements using LLMs; and (2) how to make LLM-generated features more reliable and transferable for CD-FND on unseen domains. To tackle these challenges, we propose DAUD, a novel LLM-Based Domain-Aware framework for fake news detection on Unseen Domains. DAUD employs LLMs to extract high-level semantics from news content. It models users' single- and cross-domain engagements to generate domain-aware behavioral representations. In addition, DAUD captures the relations between original data-driven features and LLM-derived features of news, users, and user engagements. This allows it to extract more reliable domain-shared representations that improve knowledge transfer to unseen domains. Extensive experiments on real-world datasets demonstrate that DAUD outperforms state-of-the-art baselines in both general and unseen-domain CD-FND settings.