Cultural adaptation in dialogue systems refers to the ability to generate responses that conform to linguistic norms and social expectations across different cultures. However, existing dialogue generation approaches pay limited attention to the role of personality in shaping cultural expression and communication style. We therefore propose HiCAP, a novel framework that integrates personality traits into culturally adaptive dialogue generation. HiCAP employs a personality-conditioned control mechanism that hierarchically modulates response style by first aligning it with the cultural tone and then refining linguistic expression and content according to individual personality attributes. To embed these factors, the model combines supervised fine-tuning and instruction-based tuning, supported by dynamic instruction templates and multi-agent demonstrations to capture diverse combinations of culture and personality. Experiments on CulturalPersonas and CPED show that HiCAP achieves the strongest overall performance across various dialogue generation metrics, demonstrating effective cultural adaptability and personality modelling.