New constraints on the physical conditions in H$_2$-bearing GRB-host damped Lyman-$\alpha$ absorbers

We report the detections of molecular hydrogen (H$_2$), vibrationally-excited H$_2$ (H$^*_2$), and neutral atomic carbon (CI), 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 $\log N$(HI, H$_2$) = $22.20\pm 0.05,~20.40\pm 0.04$ (GRB\,181020A) and $\log N$(HI, H$_2$) = $22.15\pm 0.05,~19.44\pm 0.04$ (GRB\,190114A). The DLA metallicites, depletion levels and dust extinctions are [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). We then examine the molecular gas content of all known H$_2$-bearing GRB-DLAs and explore the physical conditions and characteristics of these systems. We confirm that H$_2$ is detected in all CI- 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 $f_{\rm H_2} \gtrsim 10^{-3}$ is required for CI 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 then derive the H$_2$ excitation temperatures of the molecular gas and find that they are relatively low with $T_{\rm ex} \approx 100 - 300$\,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 the otherwise successful X-shooter GRB afterglow campaign is hampered by a significant dust bias excluding the most dust-obscured H$_2$ absorbers from identification [Abridged].