  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 $0.3\lesssim z\lesssim1.0$ 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.
