The Far Ultraviolet Spectroscopic Explorer (FUSE) has allowed precise determinations of the column densities of molecular hydrogen (H_2_) in Galactic lines of sight with a wide range of pathlengths and extinction properties. However, survey studies of lines of sight with greater extinction have been mostly restricted to the low-J states (lower total angular momentum) in which most molecular hydrogen is observed. This paper presents a survey of column densities for the molecular hydrogen in states of greater rotational excitation (J>=2) in Galactic lines of sight with log N(H_2_)>~20. This study is comprehensive through the highest excited state detectable in each line of sight. J=5 is observed in every line of sight, and we detect J=7 in four lines of sight, J=8 in one line of sight, and vibrationally excited H_2_ in two lines of sight.