Federated fine-tuning of large language models (LLMs) provides a privacy-preserving approach to deploying pervasive generative AI services, yet the substantial memory overhead of first-order (FO) gradient computation presents significant practical challenges. While zeroth-order (ZO) optimization methods offer memory-efficient alternatives, they remain susceptible to performance degradation brought by data heterogeneity. Specifically, direct ZO-for-FO substitution is incompatible with existing strategies tailored for cross-client discrepancies. In response, we propose a new federated LLM fine-tuning framework, with a holistic revamped design of the entire ZO gradient processing pipeline. Crucially, with our proposed global adaptive optimization and local personalized perturbation, we present a unified solution for incorporating ZO gradients in federated learning, from local personalized perturbation sampling and ZO gradient transmission, to global ZO gradient reconstruction and aggregation with adaptive momentum, thereby directly addressing the challenges of inefficiencies and cross-client discrepancies. Our convergence analysis and experiment results demonstrate the superiority of our proposed framework over diverse heterogeneous data settings, both in terms of generalization and efficiency.