= Solution
Let $f$ and $o$ denote the measured fundamental and first-overtone wavenumbers, and put $y=\widetilde\omega x_e$. Differences of the <Morse oscillator> levels give
$$
f=\widetilde\omega-2y,\qquad o=2\widetilde\omega-6y,
\qquad\boxed{y=f-\frac o2,\quad\widetilde\omega=3f-o.}
$$
Consequently $x_e=y/\widetilde\omega$, $D_e/(hc)=\widetilde\omega^2/(4y)$ and $D_0/(hc)=D_e/(hc)-(\widetilde\omega/2-y/4)$. Substituting the two pairs of measured bands gives the following derived quantities:
$$
\begin{array}{c|rr}
&\mathrm A&\mathrm B\\\hline
\widetilde\omega\ (\mathrm{cm^{-1}})&2648.98&2309.01\\
y\ (\mathrm{cm^{-1}})&45.22&39.64\\
x_e&0.0170707&0.0171675\\
D_e/(hc)\ (\mathrm{cm^{-1}})&38794.20&33624.67\\
E_0/(hc)\ (\mathrm{cm^{-1}})&1313.185&1144.595\\
D_0/(hc)\ (\mathrm{cm^{-1}})&37481.02&32480.07
\end{array}
$$
The conversion is $1\ \mathrm{cm^{-1}}=0.0119626566\ \mathrm{kJ\,mol^{-1}}$, after multiplication by $N_Ahc$ and consistent length units. Therefore the requested <ground-state dissociation energies> are \b[$D_0(\mathrm A)=448.4\ \mathrm{kJ\,mol^{-1}}$ and $D_0(\mathrm B)=388.5\ \mathrm{kJ\,mol^{-1}}$]. The corresponding potential-well depths are $D_e(\mathrm A)=464.1$ and $D_e(\mathrm B)=402.2\ \mathrm{kJ\,mol^{-1}}$. These are extrapolations under the stipulated <Morse oscillator> model; two low-lying bands do not establish the accuracy of the real potential near dissociation.
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