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We report the detections of molecular hydrogen (H₂), vibrationally-excited H₂ (H₂^*), and neutral atomic carbon (CI), an efficient tracer of molecular gas, in two new afterglow spectra of GRBs 181020A (z=2.938) and 190114A (z=3.376), observed with X-shooter at the Very Large Telescope (VLT). Both host-galaxy absorption systems are characterized by strong damped Lyman-α absorbers (DLAs) and substantial amounts of molecular hydrogen with logN(HI,H₂)=22.20±0.05, 20.40±0.04 (GRB 181020A) and logN(HI,H₂)=22.15±0.05, 19.44±0.04 (GRB 190114A). The DLA metallicites, depletion levels, and dust extinctions are within the typical regimes probed by GRBs with [Zn/H]=-1.57±0.06, [Zn/Fe]=0.67±0.03, and A_V=0.27±0.02mag (GRB 181020A) and [Zn/H]=-1.23±0.07, [Zn/Fe]=1.06±0.08, and A_V=0.36±0.02mag (GRB 190114A).In addition, we examine the molecular gas content of all known H₂-bearing GRB-DLAs and explore the physical conditions and characteristics required to simultaneously probe CI and H₂^*. We confirm that H₂ is detected in all CI- and H₂^*-bearing GRB absorption systems, but that these rarer features are not necessarily detected in all GRB H₂ absorbers. We find that a large molecular fraction of f_H2≥10^–3 is required for CI to be detected. The defining characteristic for H₂^* to be present is less clear, though a large H₂ column density is an essential factor. We also find that the observed line profiles of the molecular-gas tracers are kinematically "cold", with small velocity offsets of δv<20km/s from the bulk of the neutral absorbing gas. We then derive the H₂ excitation temperatures of the molecular gas and find that they are relatively low with T_ex~=100-300K, however, there could be evidence of warmer components populating the high-J H₂ levels in GRBs 181020A and 190114A. Finally, we demonstrate that even though the X-shooter GRB afterglow campaign has been successful in recovering several H₂-bearing GRB-host absorbers, this sample is still hampered by a significant dust bias excluding the most dust-obscured H₂ absorbers from identification. CI and H₂^* could open a potential route to identify molecular gas even in low-metallicity or highly dust-obscured bursts, though they are only efficient tracers for the most H₂-rich GRB-host absorption systems.