Traffic forecasting is pivotal for intelligent transportation systems, where accurate and interpretable predictions can significantly enhance operational efficiency and safety. A key challenge stems from the heterogeneity of traffic conditions across diverse locations, leading to highly varied traffic data distributions. Large language models (LLMs) show exceptional promise for few-shot learning in such dynamic and data-sparse scenarios. However, existing LLM-based solutions often rely on prompt-tuning, which can struggle to fully capture complex graph relationships and spatiotemporal dependencies-thereby limiting adaptability and interpretability in real-world traffic networks. We address these gaps by introducing Strada-LLM, a novel multivariate probabilistic forecasting LLM that explicitly models both temporal and spatial traffic patterns. By incorporating proximal traffic information as covariates, Strada-LLM more effectively captures local variations and outperforms prompt-based existing LLMs. To further enhance adaptability, we propose a lightweight distribution-derived strategy for domain adaptation, enabling parameter-efficient model updates when encountering new data distributions or altered network topologies-even under few-shot constraints. Empirical evaluations on spatio-temporal transportation datasets demonstrate that Strada-LLM consistently surpasses state-of-the-art LLM-driven and traditional GNN-based predictors. Specifically, it improves long-term forecasting by 17% in RMSE error and 16% more efficiency. Moreover, it maintains robust performance across different LLM backbones with minimal degradation, making it a versatile and powerful solution for real-world traffic prediction tasks.