Understanding the spread of narratives across multiple social media platforms is crucial for analyzing online discourse and information diffusion. This research introduces a multi-platform framework that integrates within-platform interactions and cross-layer influences to model how narratives propagate across interconnected digital ecosystems. We derive an expression for the rate of exposure and develop a five-state SEAU-D model to capture the transition of users through different stages of adoption and disengagement. To validate the model, we apply it to the spread of pro-Taiwan narratives during the 2024 Taiwan election, examining how cross-platform influence impacts the dissemination process. Our findings highlight the role of multi-platform exposure in amplifying narratives and demonstrate the significance of cross-layer influence in shaping user engagement.