In this article, we explore both numerically and analytically how the dynamical environments of mildly relativistic binaries evolve with increasing the general relativity factor γ (the normalized inverse of the binary size measured in the units of the gravitational radius corresponding to the total mass of the system). Analytically, we reveal a phenomenon of the relativistic shifting of mean-motion resonances: on increasing γ, the resonances between the test particle and the central binary shift, due to the relativistic variation of the mean motions of the primary and secondary binaries and the relativistic advance of the tertiary's pericenter. To exhibit the circumbinary dynamics globally, we numerically integrate equations of the circumbinary motion of a test particle, and construct relevant scans of the maximum Lyapunov exponents and stability diagrams in the "pericentric distance -- eccentricity" plane of initial conditions. In these scans and diagrams, regular and chaotic domains are identified straightforwardly. Our analytical and numerical estimates of the shift size are in a good agreement. Prospects for identification of the revealed effect in astronomical observations are discussed.