Let $c_j \to \infty$ be a positive sequence. Suppose there exists $s_0 \in \mathbb{R}$ such that $\sum_{j=1}^{\infty} e^{-s_0 c_j} < \infty$. I want to show that the series $\sum_{j=1}^{\infty} e^{-s c_j}$ converges for $s$ in a neighbourhood of $s_0$. Now it is easy to see that the series converges whenever $s > s_0$, but I am stuck for $s < s_0$.
I think it should be true since it works when you take $c_j = \log j$ for example (this works for $s \in (1, \infty)$, which is open).
This is not the case. If it were, that would mean that whenever $\sum_j a_j$ converges, $\sum_ja_j^\lambda$ also converges for $\lambda$ sufficiently close to $1$. (Here $a_j=\mathrm e^{-s_0c_j}$ and $\lambda=\frac s{s_0}$.) This is indeed the case for $a_j=j^{-s}$ for $s\gt1$ (your example), but not in general. For instance for $a_j=\frac1{j(\log j)^2}$, $\sum_ja_j$ converges (by the integral test), but $\sum_ja_j^\lambda$ doesn’t converge for any $\lambda\lt1$ (by comparison with $\sum_j\frac1j$).