We present new high-resolution ALMA (13CO J=1-0 and J= 2-1) and CARMA (12CO and 13CO J=1-0) observations of two Luminous Infrared Galaxies (LIRGs): Arp 55 and NGC 2623. The new data are complementary to published and archival Submillimeter Array observations of 12CO J=2-1 and J=3-2. We perform a Bayesian likelihood non-local thermodynamic equilibrium analysis to constrain the molecular gas physical conditions such as temperature, column and volume densities and the [12CO]/[13CO] abundance ratio. For Arp 55, an early/intermediate staged merger, the line measurements are consistent with cold (~10-20 K), dense (>10$^{3.5}$ cm$^{-3}$) molecular gas. For NGC 2623, the molecular gas is warmer (~110 K) and less dense (~10$^{2.7}$ cm$^{-3}$). Since Arp 55 is an early/intermediate stage merger while NGC 2623 is a merger remnant, the difference in physical conditions may be an indicator of merger stage. Comparing the temperature and volume density of several LIRGs shows that the molecular gas, averaged over ~kpc scale, of advanced mergers is in general warmer and less dense than early/intermediate stage mergers. We also find that the [12CO]/[13CO] abundance ratio of NGC 2623 is unusually high (>250) when compared to the Milky Way; however, it follows a trend seen with other LIRGs in literature. This high [12CO]/[13CO] value is very likely due to stellar nucleosynthesis enrichment of the interstellar medium. On the other hand, Arp 55 has a more Galactic [12CO]/[13CO] value with the most probable [12CO]/[13CO] value being 20-30. We measure the CO-to-H2 conversion factor, $alpha_{CO}$, to be ~0.1 and ~0.7 (3x10$^{-4}$/x$_{CO}$) M$_{odot}$ (K km s$^{-1}$ pc$^{2}$)$^{-1}$ for Arp 55 and NGC 2623, respectively. Since Arp 55 is an early/intermediate , this suggests that the transition from a Galactic conversion factor to a LIRG value happens at an even earlier merger stage.
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