I'm not sure you can hope for much. For example consider the case $c_n=1$ if $n$ is not a square, and $c_n=-1$ otherwise. The associated Dirichlet series has a pole at $s=1$, but of course the terms are not of fixed sign for sufficiently large $n$.

In general, knowing analytic properties of a Dirichlet series (such as convergence) cannot tell you much about any of the individual terms $c_n$, since you can change infinitely many of the $c_n$ for $n$ in a "sparse" set.

I'm not sure you can hope for much. For example consider the case $c_n=1$ if $n$ is not a square, and $c_n=-1$ otherwise. The associated Dirichlet series has a pole at $s=1$, but of course the terms are not of fixed sign for sufficiently large $n$.

In general, knowing analytic properties of a Dirichlet series (such as convergence) cannot tell you much about any of the individual terms $c_n$, since you can always change infinitely many of the $c_n$ for $n$ in a "sparse" set.