Social media engagement prediction is a central challenge in computational social science, particularly for understanding how users interact with misinformation. Existing approaches often treat engagement as a homogeneous time-series signal, overlooking the heterogeneous social mechanisms and platform designs that shape how misinformation spreads. In this work, we ask: ``Can neural architectures discover social exchange principles from behavioral data alone?'' We introduce \textsc{Dreams} (\underline{D}isentangled \underline{R}epresentations and \underline{E}pisodic \underline{A}daptive \underline{M}odeling for \underline{S}ocial media misinformation engagements), a social exchange theory-guided framework that models misinformation engagement as a dynamic process of social exchange. Rather than treating engagement as a static outcome, \textsc{Dreams} models it as a sequence-to-sequence adaptation problem, where each action reflects an evolving negotiation between user effort and social reward conditioned by platform context. It integrates adaptive mechanisms to learn how emotional and contextual signals propagate through time and across platforms. On a cross-platform dataset spanning $7$ platforms and 2.37M posts collected between 2021 and 2025, \textsc{Dreams} achieves state-of-the-art performance in predicting misinformation engagements, reaching a mean absolute percentage error of $19.25$\%. This is a $43.6$\% improvement over the strongest baseline. Beyond predictive gains, the model reveals consistent cross-platform patterns that align with social exchange principles, suggesting that integrating behavioral theory can enhance empirical modeling of online misinformation engagement. The source code is available at: https://github.com/ltian678/DREAMS.