Dust-obscured galaxies (DOGs) with extreme infrared luminosities may represent a key phase in the co-evolution of galaxies and supermassive black holes. We select 12 DOGs at $0.3lesssim zlesssim1.0$ with broad Mg II or H$beta$ emission lines and investigate their X-ray properties utilizing snapshot observations ($sim3~mathrm{ks}$ per source) with Chandra. By assuming that the broad lines are broadened due to virial motions of broad-line regions, we find that our sources generally have high Eddington ratios ($lambda_mathrm{Edd}$). Our sources generally have moderate intrinsic X-ray luminosities ($L_mathrm{X}lesssim10^{45}~mathrm{erg~s^{-1}}$), which are similar to those of other DOGs, but are more obscured. They also present moderate outflows and intense starbursts. Based on these findings, we conclude that high-$lambda_mathrm{Edd}$ DOGs are closer to the peaks of both host-galaxy and black-hole growth compared to other DOGs, and that AGN feedback has not swept away their reservoirs of gas. However, we cannot fully rule out the possibility that the broad lines are broadened by outflows, at least for some sources. We investigate the relations among $L_mathrm{X}$, AGN rest-frame $6~mathrm{mu m}$ monochromatic luminosity, and AGN bolometric luminosity, and find the relations are consistent with the expected ones.
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