  The star to star anticorrelation of sodium and oxygen {is} a defining feature
of globular clusters, but, to date, the astrophysical site responsible for this
unique chemical signature remains unknown. Sodium enrichment within these
clusters depends sensitively on reaction rate of the sodium destroying
reactions $^{23}$Na$(p, \gamma)$ and $^{23}$Na$(p, \alpha)$. In this paper, we
report the results of a $^{23}$Na$(^3\text{He}, d)^{24}$Mg transfer reaction
carried out at Triangle Universities Nuclear Laboratory using a $21$ MeV $^3$He
beam. Astrophysically relevant states {in $^{24}$Mg} between $11 < E_x < 12$
MeV were studied using high resolution magnetic spectroscopy, thereby allowing
the extraction of excitation energies and spectroscopic factors. Bayesian
methods are combined with the distorted wave Born approximation to assign
statistically meaningful uncertainties to the extracted spectroscopic factors.
For the first time, these uncertainties are propagated through to the
estimation of proton partial widths. Our experimental data are used to
calculate the reaction rate. The impact of the new rates are investigated using
asymptotic giant branch star models. It is found that while the astrophysical
conditions still dominate the total uncertainty, intra-model variations on
sodium production from the $^{23}$Na$(p, \gamma)$ and $^{23}$Na$(p, \alpha)$
reaction channels are a lingering source of uncertainty.
