We studied $alpha$ cluster states in $^{26}$Mg via the $^{22}$Ne($^{6}$Li,$dgamma$)$^{26}$Mg reaction in inverse kinematics at an energy of $7$ MeV/nucleon. States between $E_x$ = 4 - 12 MeV in $^{26}$Mg were populated and relative $alpha$ spectroscopic factors were determined. Some of these states correspond to resonances in the Gamow window of the $^{22}$Ne($alpha$,n)$^{25}$Mg reaction, which is one of the main neutron sources in the astrophysical $s$-process. We show that $alpha$-cluster strength of the states analyzed in this work have critical impact on s-process abundances. Using our new $^{22}$Ne($alpha$,n)$^{25}$Mg and $^{22}$Ne($alpha$,$gamma$)$^{26}$Mg reaction rates, we performed new s-process calculations for massive stars and Asymptotic Giant Branch stars and compared the resulting yields with the yields obtained using other $^{22}$Ne+$alpha$ rates from the literature. We observe an impact on the s-process abundances up to a factor of three for intermediate-mass AGB stars and up to a factor of ten for massive stars. Additionally, states in $^{25}$Mg at $E_x$ $<$ 5 MeV are identified via the $^{22}$Ne($^{6}$Li,$t$)$^{25}$Mg reaction for the first time. We present the ($^6$Li, $t$) spectroscopic factors of these states and note similarities to the $(d,p$) reaction in terms of reaction selectivity.
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