Global hydrological systems are transforming as regulation, reservoir operations, and intermittent flows reshape river regimes. Existing conceptual models, while interpretable, often underperform under conditions such as non-perennial flows and complex reservoir management. Fully distributed physically-based models, though robust, require extensive data and parameterization, limiting their utility in data-scarce contexts. Here, we develop a Hydro-Integrated Recurrent Neural Network (HIRNN), a fully differentiable hybrid framework that integrates conceptual modeling principles with the adaptive learning capabilities of recurrent neural networks. HIRNN explicitly accounts for intermittent flows and reservoir storage dynamics, achieving improved predictive performance and diagnostic insights. By explicitly modeling these key factors, HIRNN significantly improves predictive performance in 222 global basins, boosting Nash-Sutcliffe efficiency by up to 237.5\% in managed perennial catchments on modifying original model structure. This framework combines interpretability and scalability, enabling deployment in data-limited settings and informing adaptive water resource management as hydrological pressures intensify.