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We consider the two-layer moist-convective thermal rotating shallow water equations and design a flux globalization-based, well-balanced, path-conservative central-upwind numerical scheme for the studied model. We use the developed scheme to conduct a series of numerical simulations and report the observation of eastward-propagating excited dipolar systems. These systems are characterized by one or more convectively coupled, poorly isolated dipolar fronts, primarily driven by the equatorial adjustment of large-scale localized positive buoyancy or potential temperature anomalies on the equatorial beta plane. A formation of these dynamic structures is triggered when disturbances exceed a critical threshold in a moist-convective environment. Notably, during the evolution of cyclones, secondary counter-rotating anticyclones develop in the lower layer, while oppositely signed structures emerge in the upper layer, highlighting the system's vertical coupling. A significant finding of our experiments is the identification of a time lag mechanism, observable even under weaker moist-convective conditions, between the initial state and the system reaching the excited threshold required for eastward propagation. This time lag underscores a critical build-up phase, during which the system accumulates the necessary energy and momentum to transition into a dynamically active state.