We consider the Random-Cluster model on $\mathbb{Z}^d$ with interactions of infinite range of the form $J_x = ψ(x)\mathsf{e}^{-ρ(x)}$ with $ρ$ a norm on $\mathbb{Z}^d$ and $ψ$ a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of $β_{\rm sat}(s)>0$ such that the inverse correlation length in the direction $s$ is constant on $[0,β_{\rm sat}(s))$), 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 $(0,β_{\rm sat}(s))$. 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 $β$ in the regime $(β_{\rm sat}(s), β_{\rm c})$.