We report the detections of molecular hydrogen (H-2), vibrationally-excited H-2 (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-alpha absorbers (DLAs) and substantial amounts of molecular hydrogen with log N(H I, H-2) = 22.20 +/- 0.05, 20.40 +/- 0.04 (GRB 181020A) and log N(H I, H-2) = 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 H-2-bearing GRB-DLAs and explore the physical conditions and characteristics required to simultaneously probe C I and H-2*. We confirm that H-2 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 H-2 absorbers. We find that a large molecular fraction of f(H2) greater than or similar to 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 H-2 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 delta v < 20 km s(-1) from the bulk of the neutral absorbing gas. We then derive the H-2 excitation temperatures of the molecular gas and find that they are relatively low with T-ex approximate to 100-300 K, however, there could be evidence of warmer components populating the high-J H-2 levels in GRBs 181020A and 190114A. Finally, we demonstrate that even though the X-shooter GRB afterglow campaign has been successful in recovering several H-2-bearing GRB-host absorbers, this sample is still hampered by a significant dust bias excluding the most dust-obscured H-2 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 H-2-rich GRB-host absorption systems.