  The CO-to-H$_2$ conversion factor ($\alpha_\rm{CO}$) is central to measuring
the amount and properties of molecular gas. It is known to vary with
environmental conditions, and previous studies have revealed lower
$\alpha_\rm{CO}$ in the centers of some barred galaxies on kpc scales. To
unveil the physical drivers of such variations, we obtained ALMA Band 3, 6, and
7 observations toward the inner 2 kpc of NGC 3627 and NGC 4321 tracing
$^{12}$CO, $^{13}$CO, and C$^{18}$O lines on 100 pc scales. Our multi-line
modeling and Bayesian likelihood analysis of these datasets reveal variations
of molecular gas density, temperature, optical depth, and velocity dispersion,
which are among the key drivers of $\alpha_\rm{CO}$. The central 300 pc nuclei
in both galaxies show strong enhancement of temperature $T_\rm{k}>100$ K and
density $n_\rm{H_2}>10^3$ cm$^{-3}$. Assuming a CO-to-H$_2$ abundance of
$3\times10^{-4}$, we derive 4-15 times lower $\alpha_\rm{CO}$ than the Galactic
value across our maps, which agrees well with previous kpc-scale measurements.
Combining the results with our previous work on NGC 3351, we find a strong
correlation of $\alpha_\rm{CO}$ with low-J $^{12}$CO optical depths
($\tau_\rm{CO}$), as well as an anti-correlation with $T_\rm{k}$. The
$\tau_\rm{CO}$ correlation explains most of the $\alpha_\rm{CO}$ variation in
the three galaxy centers, whereas changes in $T_\rm{k}$ influence
$\alpha_\rm{CO}$ to second order. Overall, the observed line width and
$^{12}$CO/$^{13}$CO 2-1 line ratio correlate with $\tau_\rm{CO}$ variation in
these centers, and thus they are useful observational indicators for
$\alpha_\rm{CO}$ variation. We also test current simulation-based
$\alpha_\rm{CO}$ prescriptions and find a systematic overprediction, which
likely originates from the mismatch of gas conditions between our data and the
simulations.
