We study the near-critical FK-Ising model. First, a determination of the correlation length defined via crossing probabilities is provided. Second, a striking phenomenon about the near-critical behavior of FK-Ising is highlighted, which is completely missing from the case of standard percolation: in any monotone coupling of FK configurations $omega_p$ (e.g., in the one introduced in [Gri95]), as one raises $p$ near $p_c$, the new edges arrive in a fascinating self-organized way, so that the correlation length is not governed anymore by the amount of pivotal edges at criticality. In particular, it is smaller than the heat-bath dynamical correlation length determined in the forthcoming [GP]. We also include a discussion of near-critical and dynamical regimes for general random-cluster models. For the heat-bath dynamics in critical random-cluster models, we conjecture that there is a regime of values of cluster-weights $q$ where there exist macroscopic pivotals yet there are no exceptional times. These are the first natural models that are expected to be noise sensitive but not dynamically sensitive.