  In this work, we have studied the $2\nu\beta\beta$ decay of $^{76}$Ge and
$^{96}$Zr isotopes utilizing large-scale shell-model calculations. The GWBXG
effective interaction has been employed in the calculation of
$2\nu\beta\beta$-decay nuclear matrix elements (NMEs). We have tested the
effective interaction by comparing the predicted spectroscopic properties, such
as energy spectra and transition probabilities, with the available experimental
data. The variation of cumulative NMEs with respect to the $1^+$ state energies
of the intermediate nucleus is also studied, corresponding to $0^+_{\rm
g.s}\rightarrow0^+_{\rm g.s}$, $0^+_{\rm g.s}\rightarrow0^+_{2}$, and $0^+_{\rm
g.s}\rightarrow2^+_{1}$ transitions between the parent and granddaughter
nuclei. The extracted half-lives using the shell-model predicted NMEs show good
agreement with the recent experimental data. The comparison of the shell-model
predicted NMEs with previous NMEs available in the literature is discussed.
Also, the computed branching ratios for the $2\nu\beta\beta$ decay of $^{76}$Ge
and both the $2\nu\beta\beta$ and single-$\beta$ decay of $^{96}$Zr are
reported.
