It has recently been demonstrated that dynamical magnetic correlations measured by neutron scattering in iron chalcogenides can be described with models of short-range correlations characterized by particular {choices of four-spin plaquettes, where the appropriate choice changes as the} parent material is doped towards superconductivity. Here we apply such models to describe measured maps of magnetic scattering as a function of two-dimensional wave vectors obtained for optimally superconducting crystals of FeTe$_{1-x}$Se$_x$. We show that the characteristic antiferromagnetic wave vector evolves from that of the bicollinear structure found in underdoped chalcogenides (at high temperature) to that associated with the stripe structure of antiferromagnetic iron arsenides (at low temperature); {these can both be described with the same local plaquette, but with different inter-plaquette correlations}. While the magnitude of the low-energy magnetic spectral weight is substantial at all temperatures, it actually weakens somewhat at low temperature, where the charge carriers become more itinerant. The observed change in spin correlations is correlated with the dramatic drop in the electronic scattering rate and the growth of the bulk nematic response on cooling. Finally, we also present powder neutron diffraction results for lattice parameters in FeTe$_{1-x}$Se$_x$ indicating that the tetrahedral bond angle tends to increase towards the ideal value on cooling, in agreement with the increased screening of the crystal field by more itinerant electrons and the correspondingly smaller splitting of the Fe $3d$ orbitals.
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