We consider the Random-Cluster model on $\mathbb{Z}^d$ with interactions of
infinite range of the form $J_x = \psi(x)\mathsf{e}^{-\rho(x)}$ with $\rho$ a
norm on $\mathbb{Z}^d$ and $\psi$ a subexponential correction. We first provide
an optimal criterion ensuring the existence of a nontrivial saturation regime
(that is, the existence of $\beta_{\rm sat}(s)>0$ such that the inverse
correlation length in the direction $s$ is constant on $[0,\beta_{\rm
sat}(s))$), thus removing a regularity assumption used in a previous work of
ours. Then, under suitable assumptions, we derive sharp asymptotics (which are
not of Ornstein-Zernike form) for the two-point function in the whole
saturation regime $(0,\beta_{\rm sat}(s))$. We also obtain a number of
additional results for this class of models, including sharpness of the phase
transition, mixing above the critical temperature and the strict monotonicity
of the inverse correlation length in $\beta$ in the regime $(\beta_{\rm
sat}(s), \beta_{\rm c})$.
