Direct numerical simulations of non-premixed fuel-rich methane-oxygen flames at 20 bar are conducted to investigate the turbulent mixing burning of gaseous propellants in rocket engines. The reacting flow is simulated using the EBI-DNS solver, within the OpenFOAM frame. The transport of species is resolved with finite rate chemistry, using a complex skeletal mechanism that entails 21 species. Two different flames at low and high Reynolds numbers are considered to study the sensitivity of the flame dynamics to turbulence. Regime markers are used to measure the probability of the flow to burning in premixed and non-premixed conditions at different regions. The local heat release statistics are studied to understand the drivers in the development of the turbulent diffusion flame. Despite the eminent non-premixed configuration, a significant amount of combustion takes place in premixed conditions. Premixed combustion is viable in both lean and fuel-rich regions, relatively far from the stoichiometric line. It is found that a growing turbulent kinetic energy is detrimental to combustion in fuel-rich premixed conditions. This is motivated by the disruption of the local premixed flame front, which promotes fuel transport into the diffusion flame. In addition, at downstream positions, higher turbulence enables the advection of methane into the lean core of the flame, enhancing the burning rates in these regions. Hence, the primary effect of turbulence is to increase the fraction of propellants burnt in oxygen-rich and near stoichiometric conditions. As a consequence, the mixture fraction of the products shifts towards lean conditions, influencing combustion completion at downstream positions.