Adaptive voter models (AVMs) are simple mechanistic systems that model the emergence of mesoscopic structure from local networked processes driven by conflict and homophily. AVMs display rich behavior, including a phase transition from a fully-fragmented regime of "echo-chambers" to a regime of persistent disagreement governed by low-dimensional quasistable manifolds. Many extant methods for approximating the behavior of AVMs are either restricted in scope, expensive in computation, or inaccurate in predicting important statistics. In this work, we develop a novel, second-order moment closure approximation method for binary-state rewire-to-random and rewire-to-same model variants. We incorporate a small amount of noise via a random mutation term, which renders the system ergodic. Using ergodicity, we then approximate the voting process, which is non-Markovian in the second moments of the system, with a Markovian term near the phase transition. This approximation exploits an asymmetry between different classes of voting events. The resulting scheme enables us to predict the location of the phase transition and the active edge density in the regime of persistent disagreement, across the entire space of parameters and opinion densities. Numerically, our results are nearly exact for the rewire-to-random model, and competitive with other current approaches for the rewire-to-same model. Moreover, our computations display constant scaling in the mean degree, enabling approximations for denser systems than previously possible. We conclude with suggestions for model refinements and extensions.