  The {\it Far Ultraviolet Spectroscopic Explorer} ({\it FUSE}) has allowed
precise determinations of the column densities of molecular hydrogen ($\Hmol$)
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 ($J \geq 2$) in Galactic lines of sight with
$\log{\NHmol} \gtrsim 20$. 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 $\Hmol$ in two lines of sight. We compared the
apparent $b$-values and velocity offsets of the higher-$J$ states relative to
the dominant low-$J$ states and we found no evidence of any trends that might
provide insight into the formation of higher-$J$ $\Hmol$, although these
results are the most affected by the limits of the {\it FUSE} resolution. We
also derive excitation temperatures based on the column densities of the
different states. We confirm that at least two distinct temperatures are
necessary to adequately describe these lines of sight, and that more
temperatures are probably necessary. Total $\Hmol$ column density is known to
be correlated with other molecules; we explore if correlations vary as a
function of $J$ for several molecules, most importantly CH and CH$^+$. Finally,
we briefly discuss interpretations of selected lines of sight by comparing them
to models computed using the Meudon PDR code.
