Accurate time integration of hyperbolic-parabolic systems, particularly in the presence of shocks and steep gradients, remains a central challenge in computational fluid dynamics. In this work, we propose a robust, adaptive time integration framework for discontinuous Galerkin discretizations that combines an embedded third-order Strong Stability Preserving Runge-Kutta method with physics-based shock capturing and novel error control strategies. The proposed method is based on total variation diminishing properties while leveraging a proportional-integral controller for adaptive step-size selection, eliminating the need for empirical CFL tuning. A key innovation lies in the introduction of an entropy-based filtering mechanism that modulates element-wise error estimates, effectively dampening spurious spikes induced by discontinuities. Additionally, the integral term of the PI controller is stabilized using a moving median over a sliding window, enhancing reliability in shock-dominated regimes. The overall methodology requires no parameter tuning beyond a user-defined error tolerance (as it is common in any ordinary differential equation) and is demonstrated to be stable and accurate across a broad range of canonical test cases. Compared to conventional CFL stable solution obtained for the same numerical setups in a previous work, the proposed approach consistently delivers improved accuracy and robustness for high-fidelity simulations in complex compressible flows.