This is not true without additional integrability conditions on $f(\cdot+iy)$.

$\hat{f}(w)=o(e^{-a|w|})$ implies that $e^{b|w|}\hat{f}(w)\in L_2(\mathbb{R})$ for all $b < a$. The latter inclusion holds if and only if $f(z)$ is analytic in the strip $|\Im(z)|\leq a$ and $$ \sup\limits_{|y|\leq b}\|f(\cdot+iy)\|_{L^2(\mathbb R)}<\infty$$ for all $b< a$ (see Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2, Theorem IX.13)).

This is not true without additional integrability conditions on $f(\cdot+iy)$.

$\hat{f}(w)=o(e^{-a|w|})$ implies that $e^{b|w|}\hat{f}(w)\in L_2(\mathbb{R})$ for all $b < a$. The latter inclusion holds if and only if $f(z)$ is analytic in the strip $|\Im(z)|< a$ and $$ \sup\limits_{|y|\leq b}\|f(\cdot+iy)\|_{L^2(\mathbb R)}<\infty$$ for all $b< a$ (see Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2, Theorem IX.13)).

This is not true without additional integrability conditions on $f(\cdot+iy)$.

$\hat{f}(w)=o(e^{-a|w|})$ implies that $e^{b|w|}\hat{f}(w)\in L_2(\mathbb{R})$ for all $b < a$. The latter inclusion holds iff $f(z)$ is analytic in the strip $|\Im(z)|\leq a$ and  $$ \sup\limits_{|y|\leq b}\|f(\cdot+iy)\|_{L^2(\mathbb R)}<\infty$$ for all $b< a$ (see Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2, Theorem IX.13)).

This is not true without additional integrability conditions on $f(\cdot+iy)$.

$\hat{f}(w)=o(e^{-a|w|})$ implies that $e^{b|w|}\hat{f}(w)\in L_2(\mathbb{R})$ for all $b < a$. The latter inclusion holds if and only if $f(z)$ is analytic in the strip $|\Im(z)|\leq a$ and  $$ \sup\limits_{|y|\leq b}\|f(\cdot+iy)\|_{L^2(\mathbb R)}<\infty$$ for all $b< a$ (see Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2, Theorem IX.13)).