Zero-shot object-goal navigation (ZSON) is a challenging problem in robotics that requires a comprehensive understanding of both language and visual observations. Contextual cues from rooms and objects are critical, but their relative importance depends on the target: some objects are strongly tied to specific room types, while others are better predicted by nearby co-located objects. Existing methods overlook this distinction, leading to inefficient and inaccurate exploration. We present CLUE, a novel navigation framework that adaptively balances the use of contextual rooms and objects by leveraging commonsense knowledge extracted from an offline large language model (LLM). By estimating a target's association with room types using LLM, the agent prioritizes room cues for predictable objects and object cues for those with weak room associations. Our framework constructs a unified semantic value map that integrates both types of contextual information, adaptively weighted by the target's ambiguity to guide exploration. Combined with multi-viewpoint verification and an exploration strategy informed by contextual cues, CLUE achieves robust and efficient navigation. Extensive experiments in simulation and real-world deployments show that our method consistently outperforms state-of-the-art baselines in both success rate (SR) and success weighted by path length (SPL), demonstrating its effectiveness and practicality for real-world navigation tasks.