  In a rapidly changing shape phase region, the presence of shape coexistence
and its possible impact on the decay modes and half$-$lives, has been explored
in astrophysically interesting Mo and Ru isotopes, in an extensive study within
the microscopic theoretical framework using Nilsson Strutinsky Method and
Relativistic Mean Field Model. The isotopic chains of Mo and Ru exhibit rapid
shape phase transitions, triaxial $\gamma$ softness, shape instability along
with many coexisting states mostly with oblate and triaxial shapes. Proton and
neutron separation energies have been calculated and compared with the
available data. Results obtained from both the formalisms are in good agreement
with each other as well as the available experimental data. Our computed
$\beta-$decay half$-$lives and separation energy for nuclei exhibiting shape
coexistence were examined for the decay mode from second minima state of the
parent nuclei to the ground or excited state of the daughter nuclei. The second
minima state of the coexisting shapes in Mo and Ru isotopes, were seen to
impact the structural properties, $\beta-$decay half$-$lives, separation energy
and hence the stability of the nuclei.
