Fine-tuning large language models (LLMs) in federated settings enables privacy-preserving adaptation but suffers from cross-client interference due to model aggregation. Existing federated LoRA fine-tuning methods, primarily based on FedAvg, struggle with data heterogeneity, leading to harmful cross-client interference and suboptimal personalization. In this work, we propose \textbf{FedALT}, a novel personalized federated LoRA fine-tuning algorithm that fundamentally departs from FedAvg. Instead of using an aggregated model to initialize local training, each client continues training its individual LoRA while incorporating shared knowledge through a separate Rest-of-World (RoW) LoRA component. To effectively balance local adaptation and global information, FedALT introduces an adaptive mixer that dynamically learns input-specific weightings between the individual and RoW LoRA components, drawing conceptual foundations from the Mixture-of-Experts (MoE) paradigm. Through extensive experiments on NLP benchmarks, we demonstrate that FedALT significantly outperforms state-of-the-art personalized federated LoRA fine-tuning methods, achieving superior local adaptation without sacrificing computational efficiency.