Physical problems involving strong shock advection, such as hypersonic flows and detonation phenomena, pose significant challenges and are well known to be ill suited for conventional model order reduction (MOR) techniques. This limitation primarily stems from the slow decay of the Kolmogorov N-width in shock-dominated regimes. Adaptive reduced-order models (ROMs) have recently emerged as a promising alternative by dynamically updating reduced bases and sampling locations during online simulations, enabling accurate tracking of evolving localized dynamics. However, the frequent basis and sampling updates required by adaptive ROMs can substantially reduce their computational advantage relative to static ROM approaches. In this work, we introduce a self-adaptive reduced-order modeling framework that dynamically selects and deploys the most appropriate ROM—either adaptive or static—during online execution. The selection is guided by rigorous multilevel a priori and a posteriori error estimators, enabling an optimal balance between computational efficiency and modeling accuracy while preserving fidelity to the local shock physics. For the static ROM component, we further investigate several nonlinear reduced-basis strategies designed to mitigate the Kolmogorov barrier in shock-dominated flows. The proposed framework is evaluated on two representative shock-driven problems: (1) a pseudo-one-dimensional rotating detonation wave, and (2) a two-dimensional rotating detonation engine configuration.