In the rapidly evolving field of artificial intelligence (AI), Generative AI (GAI) has become a crucial driver for enhancing organizational agility and operational efficiency. This study explores the complex interplay between awareness, digital technology adoption, and infrastructure, and their collective influence on both internal and external GAI agility and customer engagement. Utilizing Partial Least Squares Structural Equation Modeling (PLS-SEM), the research analyzed data from 217 participants across diverse industries, including manufacturing, construction, information and communications, retail, law, medical and health services, education, and services. The results revealed that both awareness and infrastructure play a crucial role in boosting internal agility within GAI systems, signifying that a deep understanding and a robust setup are key drivers for effective internal AI operations. Interestingly, when combined with digital technology adoption, these factors also positively influenced external GAI agility, indicating that embracing new technologies enhances an organization's adaptability in external interactions. However, digital technology adoption alone did not significantly alter internal GAI agility, suggesting that other elements might be at play in the internal dynamics of GAI. Furthermore, internal GAI agility appeared to have only a modest effect on enhancing customer engagement, highlighting a potential gap between internal AI efficiencies and tangible customer-facing outcomes. These insights are critical in understanding the determinants of GAI agility and its role in improving customer engagement across various sectors. The study offers valuable contributions to the theoretical and practical understanding of GAI, emphasizing the importance of strategic awareness, technology adoption, and infrastructure development in leveraging GAI's benefits in diverse business environments.