The first thing to get straight is that convergence of the Fourier transform of $f$ does not imply absolute integrability, i.e., that $f \in L^1(\mathbb{R})$.
For a counterexample, take $f(x) = \sin (x^2)$ where
$$\int_{-\infty}^\infty\sin(x^2)e^{ikx} \, dx = 2\int_0^\infty\sin(x^2) \cos(kx) \, dx = \int_0^\infty(\sin(x^2+kx)+ \sin(x^2-kx)) \, dx$$
Making the change of variables $x = u \mp \frac{k}{2}$ we get
$$\begin{align} \int_0^\infty \sin(x^2\pm kx)\, dx &= \int_0^\infty \sin\left(\left(u\mp \frac{k}{2}\right)^2 \pm k\left(u \mp \frac{k}{2}\right)\right)\, du \\ &= \int_0^\infty\sin\left(u^2 - \frac{k^2}{2}\right)\, du \\ &= \cos \left(\frac{k^2}{4}\right)\int_0^\infty \sin (u^2) \, du - \sin \left(\frac{k^2}{4}\right)\int_0^\infty \cos (u^2) \, du \end{align}$$
The well-known Fresnel integrals on the RHS are $\displaystyle \int_0^\infty \cos (u^2) \, du= \int_0^\infty \sin (u^2) \, du = \sqrt{\frac{\pi}{8}}$.
Substituting above, we get
$$\hat{f}(k) = \int_{-\infty}^\infty\sin(x^2)e^{ikx} \, dx = \sqrt{\frac{\pi}{2}}\left( \cos \left(\frac{k^2}{4}\right)-\sin \left(\frac{k^2}{4}\right)\right), $$
and the Fourier transform converges. However, $f \not\in L^1(\mathbb{R})$. This can either be shown directly or inferred from the fact that
$$\lim_{k \to \infty} \left( \cos \left(\frac{k^2}{4}\right)-\sin \left(\frac{k^2}{4}\right)\right)\,\, \text{DNE},$$
If $f \in L^1(\mathbb{R})$ then by the Riemann-Lebesgue lemma we would have $\lim_{k \to \infty}\hat{f}(k) = 0$
ROC Proof
Suppose $x$ is locally integrable and for some $s_0 \in \mathbb{C}$ we have convergence of the unilateral Laplace transform
$$X_+(s_0) = \int_0^\infty x(t) e^{-s_0t} \, dt$$
Let $\alpha(t) = \int_0^t x(u)e^{-s_0 u} \, du$. For any $T>0$ it follows from a basic property of the Riemann-Stieltjes integral (proved here) that
$$\int_0^T x(t) e^{-st}\, dt = \int_0^T e^{-(s-s_0)t} x(t) e^{-s_0t}\, dt = \int_0^T e^{-(s-s_0)t} \, d\alpha(t)$$
Integrating by parts on the RHS, we get
$$\tag{*}\int_0^T x(t) e^{-st}\, dt = \alpha(T)e^{-(s-s_0)T} + (s-s_0)\int_0^T e^{-(s-s_0)t} \alpha(t) \, dt$$
Since $X_+(s_0)$ converges, it follows that $|\alpha(t)|$ is bounded for all $t \geqslant 0$. This implies that for all $s$ such that $\Re(s) > \Re(s_0)$, we have
$$\lim_{T \to \infty}\alpha(T)e^{-(s-s_0)T} = 0,$$
and the integral on the RHS of (*) is absolutely convergent.
Thus,
$$X_+(s) = \lim_{T \to \infty}\int_0^T x(t) e^{-st}\, dt= \lim_{T\to \infty}\int_0^T e^{-(s-s_0)t} \, d\alpha(t) = \int_0^\infty e^{-(s-s_0)t} \, d\alpha(t),$$
where $X_+(s)$ converges for all $s$ such that $\Re(s) > \Re(s_0)$.
In a similar way we can show that if we have convergence of
$$X_-(s_1) = \int_{-\infty}^0 x(t) e^{-s_1t} \, dt,$$
then $X_-(s)$ converges for all $s$ such that $\Re(s) < \Re(s_1)$.
Best Answer
You are right that the straightforward comparison $$|e^{t-zt} \sin(e^t)|\le e^{t(1-\operatorname{Re}z)}$$ only yields a conclusive result for $\operatorname{Re}z>1$. But let's integrate by parts: $$ \int_1^A e^{t-zt} \sin(e^t)\,dt = -e^{-zt} \cos(e^t)\bigg|_1^A - z\int_1^A e^{-zt} \cos(e^t)\,du $$ Now use $|e^{-zt} \cos(e^t)|\le e^{-t \operatorname{Re}z }$.
As $A\to\infty$, both terms have finite limits provided $\operatorname{Re}z>0$.