We study $W^+W^-$ and Zh final sstates at future linear $e^+e^-$ colliders; designing analyses specific to the various final state polarizations allows us to target specific beyond the Standard Model (BSM) effects, parametrized in the form of dimension-6 operators. We find that CLIC can access effects roughly an order of magnitude smaller than HL-LHC or ILC, and two orders of magnitude smaller than LEP. These results are interpreted in the context of well-motivated BSM scenarios–at weak and strong coupling–where we expect correlated effects in Drell–Yann processes. The latter turn out to have better discovery potential, although the diboson processes provide additional discriminating power, potentially furnishing a way to measure the spin and coupling of BSM states.