Recently, astronomers have found a population of dust-obscured galaxies (DOGs) with high infrared (IR) luminosities, some of which may represent a key phase in the co-evolution framework of galaxies and supermassive black holes (SMBHs). In this work, we select 12 DOGs with broad Mg II or Hβ emission lines and analyze their X-ray properties utilizing snapshot observations (~ 3 ks per source) with Chandra. We find that they generally have moderate intrinsic X-ray luminosities (LX), which are similar to other DOGs, but are more obscured. Our sources also present moderately strong outflows and intense starbursts. We argue that these facts can be well-explained if we assume that the Mg II or Hβ lines are broadened due to virial motions of broad-line regions (BLRs). Under this assumption, we find that our sources generally have high Eddington ratios (lEdd), and conclude that high-lEdd 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, it is also possible that the broad lines are broadened by outflows, at least for some sources. We investigate the relations among LX, AGN rest-frame 6 um monochromatic luminosity (L6 um), and AGN bolometric luminosity (Lbol), and find the relations are consistent with the expected ones. There is also tentative evidence indicating that the [O III]-based outflow strength is positively correlated with both LX and column density (NH).