We consider FK percolation on Z^d with interactions of infinite range of the form J_x = ψ(x) exp(-ρ(x)) with ρ a norm on 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 ꞵ_sat(s) > 0 such that the inverse correlation length in the direction s is constant on [0, ꞵ_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, ꞵ_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 ꞵ in the regime (ꞵ_sat(s), ꞵ_c).