  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.
