Safety evaluation of Large Language Models (LLMs) has made progress and attracted academic interest, but it remains challenging to keep pace with the rapid integration of LLMs across diverse applications. Different applications expose users to various harms, necessitating application-specific safety evaluations with tailored harms and policies. Another major gap is the lack of focus on the dynamic and conversational nature of LLM systems. Such potential oversights can lead to harms that go unnoticed in standard safety benchmarks. This paper identifies the above as key requirements for robust LLM safety evaluation and recognizing that current evaluation methodologies do not satisfy these, we introduce the textttSAGE (Safety AI Generic Evaluation) framework. textttSAGE is an automated modular framework designed for customized and dynamic harm evaluations. It utilizes adversarial user models that are system-aware and have unique personalities, enabling a holistic red-teaming evaluation. We demonstrate textttSAGE's effectiveness by evaluating seven state-of-the-art LLMs across three applications and harm policies. Our experiments with multi-turn conversational evaluations revealed a concerning finding that harm steadily increases with conversation length. Furthermore, we observe significant disparities in model behavior when exposed to different user personalities and scenarios. Our findings also reveal that some models minimize harmful outputs by employing severe refusal tactics that can hinder their usefulness. These insights highlight the necessity of adaptive and context-specific testing to ensure better safety alignment and safer deployment of LLMs in real-world scenarios.