Recent advances in Visual Question Answering (VQA) have demonstrated impressive performance in natural image domains, with models like LLaVA leveraging large language models (LLMs) for open-ended reasoning. However, their generalization degrades significantly when transferred to out-of-domain scenarios such as remote sensing, medical imaging, or math diagrams, due to large distributional shifts and the lack of effective domain adaptation mechanisms. Existing approaches typically rely on per-domain fine-tuning or bespoke pipelines, which are costly, inflexible, and not scalable across diverse tasks. In this paper, we propose CATCH, a plug-and-play framework for cross-domain adaptation that improves the generalization of VQA models while requiring minimal changes to their core architecture. Our key idea is to decouple visual and linguistic adaptation by introducing two lightweight modules: a domain classifier to identify the input image type, and a dual adapter mechanism comprising a Prompt Adapter for language modulation and a Visual Adapter for vision feature adjustment. Both modules are dynamically injected via a unified hook interface, requiring no retraining of the backbone model. Experimental results across four domain-specific VQA benchmarks demonstrate that our framework achieves consistent performance gains without retraining the backbone model, including +2.3 BLEU on MathVQA, +2.6 VQA on MedVQA-RAD, and +3.1 ROUGE on ChartQA. These results highlight that CATCH provides a scalable and extensible approach to multi-domain VQA, enabling practical deployment across diverse application domains.