Large Language Models (LLMs) have demonstrated impressive capabilities on a broad array of NLP tasks, but their tendency to produce hallucinations$\unicode{x2013}$plausible-sounding but factually incorrect content$\unicode{x2013}$poses severe challenges in high-stakes domains. Existing hallucination detection methods either bear the computational cost of multiple inference passes or sacrifice accuracy for efficiency with single-pass approaches, neither of which is ideal in resource-constrained environments such as edge devices. We propose the Shannon Entropy Distribution Hallucination Detector (ShED-HD), a novel hallucination detection framework that bridges this gap by classifying sequence-level entropy patterns using a lightweight BiLSTM architecture with single-headed attention. In contrast to prior approaches, ShED-HD efficiently detects distinctive uncertainty patterns across entire output sequences, preserving contextual awareness. Through in-depth evaluation on three datasets (BioASQ, TriviaQA, and Jeopardy Questions), we show that ShED-HD significantly outperforms other computationally efficient approaches in the out-of-distribution setting, while achieving comparable performance in the in-distribution setting. ShED-HD facilitates hallucination detection that is low-cost, accurate, and generalizable, improving the credibility of content generated by LLMs in resource-constrained environments where trustworthy AI functionality is crucial.