We report the detections of molecular hydrogen (H₂), vibrationally-excited H₂ (H ₂^∗ ), and neutral atomic carbon (C I), 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(H I, H₂) = 22.20 ± 0.05, 20.40 ± 0.04 (GRB 181020A) and logN(H I, 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 ± 0.02 mag (GRB 181020A) and [Zn/H] = −1.23 ± 0.07, [Zn/Fe] = 1.06 ± 0.08, and A_V = 0.36 ± 0.02 mag (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 C I and H ₂^∗ . We confirm that H₂ is detected in all C I- and H ₂^∗ -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 C I to be detected. The defining characteristic for H ₂^∗ 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 < 20 km s⁻¹ 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 ≈ 100−300 K, 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. C I and H ₂^∗ 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.