Due to their low-complexity and energy-efficiency, unified simultaneous wireless information and power transfer (U-SWIPT) receivers are especially suitable for low-power Internet of Things (IoT) applications. Towards accurately modeling practical operating conditions, in this study, we provide a unified transient framework for a dual-diode U-SWIPT that jointly accounts for diode nonlinearity and capacitor-induced memory effects. The proposed model accurately describes the inherent time dependence of the rectifier, highlighting its fundamental impact on both energy harvesting (EH) and information decoding (ID) processes. Based on the provided memory-aware model, we design a low-complexity adaptive detector that learns the nonlinear state transition dynamics and performs decision-directed detection with linear complexity. The proposed detection scheme approaches maximum likelihood sequence detection (MLSD) performance in memory-dominated regimes, while avoiding the exponential search required by classical sequence detection. Overall, these results demonstrate that properly exploiting rectifier memory provides a better tradeoff between data rate and reliability for U-SWIPT receivers.