We present a study of the molecular gas in early-mid stage major mergers, with a sample of 43 major-merger galaxy pairs selected from the Mapping Nearby Galaxies at Apache Point Observatory survey and a control sample of 195 isolated galaxies selected from the xCOLD GASS survey. Adopting kinematic asymmetry as a new effective indicator to describe the merger stage, we aim to study the role of molecular gas in the merger-induced star formation enhancement along the merger sequence of galaxy pairs. We obtain the molecular gas properties from CO observations with the James Clerk Maxwell Telescope, Institut de Radioastronomie Milimetrique 30 m telescope, and the MaNGA-ARO Survey of CO Targets survey. Using these data, we investigate the differences in molecular gas fraction ( fH2 ), star formation rate (SFR), star formation efficiency (SFE), molecular-to-atomic gas ratio ( MH2 /M H i ), total gas fraction (f gas), and the SFE of total gas (SFEgas) between the pair and control samples. In the full pair sample, our results suggest the fH2 of paired galaxies is significantly enhanced, while the SFE is comparable to that of isolated galaxies. We detect significantly increased fH2 and MH2 /M H i in paired galaxies at the pericenter stage, indicating an accelerated transition from atomic gas to molecular gas due to interactions. Our results indicate that the elevation of fH2 plays a major role in the enhancement of global SFR in paired galaxies at the pericenter stage, while the contribution of enhanced SFE in specific regions requires further explorations through spatially resolved observations of a larger sample spanning a wide range of merger stages.