  Investigating the molecular gas in the inner regions of protoplanetary disks
provides insight into how the molecular disk environment changes during the
transition from primordial to debris disk systems. We conduct a small survey of
molecular hydrogen (H$_2$) fluorescent emission, using 14 well-studied
Classical T Tauri stars at two distinct dust disk evolutionary stages, to
explore how the structure of the inner molecular disk changes as the optically
thick warm dust dissipates. We simulate the observed HI-Lyman $\alpha$-pumped
H$_2$ disk fluorescence by creating a 2D radiative transfer model that
describes the radial distributions of H$_{2}$ emission in the disk atmosphere
and compare these to observations from the Hubble Space Telescope. We find the
radial distributions that best describe the observed H$_2$ FUV emission arising
in primordial disk targets (full dust disk) are demonstrably different than
those of transition disks (little-to-no warm dust observed). For each best-fit
model, we estimate inner and outer disk emission boundaries (r$_{in}$ and
r$_{out}$), describing where the bulk of the observed H$_2$ emission arises in
each disk, and we examine correlations between these and several observational
disk evolution indicators, such as n$_{13-31}$, r$_{in,CO}$, and the mass
accretion rate. We find strong, positive correlations between the H$_2$ radial
distributions and the slope of the dust SED, implying the behavior of the
molecular disk atmosphere changes as the inner dust clears in evolving
protoplanetary disks. Overall, we find that H$_2$ inner radii are $\sim$4 times
larger in transition systems, while the bulk of the H$_2$ emission originates
inside the dust gap radius for all transitional sources.
