This paper introduces a distributed operational solution for integrated transmission-distribution (ITD) system management. A fundamental challenge in designing distributed approaches for AC optimal power flow (OPF) problems in ITD systems is the nonconvexity and nonlinearity of the optimization problems for both transmission and distribution systems. To tackle the challenges, our research introduces an enhanced version of the Augmented Lagrangian based Alternating Direction Inexact Newton method (aladin), which incorporates a second-order correction strategy and convexification. The former improves the algorithm's ability to follow curved trajectories effectively with minimal additional computational demand, while the latter simplifies the decoupled subproblems without introducing the combinatory complexity typically associated with additional inequality constraints. The theoretical studies demonstrate that the proposed distributed algorithm operates the ITD systems with a local quadratic convergence guarantee. Extensive simulations on various ITD configurations highlight the superior performance of our distributed approach in terms of convergence speed, computational efficiency, scalability, and adaptability.