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 \pm 0.05 ,~ { } 20.40 \pm 0.04 ( GRB 181020A ) and \log N ( H i , H _ { 2 } ) = 22.15 \pm 0.05 ,~ { } 19.44 \pm 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 \pm 0.06 , [ Zn/Fe ] = 0.67 \pm 0.03 , and A _ { V } = 0.27 \pm 0.02 mag ( GRB 181020A ) and [ Zn/H ] = -1.23 \pm 0.07 , [ Zn/Fe ] = 1.06 \pm 0.08 , and A _ { V } = 0.36 \pm 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 _ { H _ { 2 } } \gtrsim 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 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 } \approx 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 .