Large language models (LLMs) commonly risk copyright infringement by reproducing protected content verbatim or with insufficient transformative modifications, posing significant ethical, legal, and practical concerns. Current inference-time safeguards predominantly rely on restrictive refusal-based filters, often compromising the practical utility of these models. To address this, we collaborated closely with intellectual property experts to develop LAW-LM (Legally Aware Language Model), a legally-grounded framework explicitly designed to align LLM outputs with fair-use doctrine. Central to our method is FairUseDB, a carefully constructed dataset containing 18,000 expert-validated examples covering nine realistic infringement scenarios. Leveraging this dataset, we apply Direct Preference Optimization (DPO) to fine-tune open-source LLMs, encouraging them to produce legally compliant and practically useful alternatives rather than resorting to blunt refusal. Recognizing the shortcomings of traditional evaluation metrics, we propose new measures: Weighted Penalty Utility and Compliance Aware Harmonic Mean (CAH) to balance infringement risk against response utility. Extensive quantitative experiments coupled with expert evaluations confirm that LAW-LM substantially reduces problematic outputs compared to state-of-the-art approaches, while preserving real-world usability.