Large Language Model (LLM) factual hallucination critically limits their reliability. To address this, we introduce FactPrompt, a framework inspired by cognitive refinement processes, integrating guided Chain-of-Thought (CoT) with a novel factored self-correction mechanism. FactPrompt first generates an initial response. Crucially, it then deliberately decouples verification: it independently generates internal factual claims relevant to the query using a distinct prompt, separate from the initial reasoning path. This structural separation allows the LLM to evaluate the initial response against these independently sourced claims, minimizing bias propagation from potentially flawed initial reasoning—a key distinction from standard CoT or simpler self-critique loops. Concurrently, we propose and validate R&T, an information-theoretic metric quantifying deviations from uniform information density (UID) that signal potential unreliability. R&T strongly correlates with established accuracy metrics, supporting its use for automated, ground-truth-free reliability assessment. Extensive evaluation across ten datasets shows FactPrompt yields substantial accuracy improvements over strong baselines (average 4.3%, up to 8.37%), notably in long-form generation and open-ended QA. FactPrompt demonstrates that structured, decoupled internal verification is a promising route towards more reliable and trustworthy LLMs.