This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $β$, $γ$, $δ$, $η$, $ν$, $ζ$ as well as $α$ (when $α\ge0$). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalization of Kesten's classical scaling relation for Bernoulli percolation involving the ``mixing rate'' critical exponent $ι$ replacing the four-arm event exponent $ξ_4$.