Context. The late evolutionary phase of low and intermediate-mass stars is strongly constrained by their mass-loss rate, which is orders of magnitude higher than during the main sequence. The wind surrounding these cool expanded stars frequently shows non-spherical symmetry, thought to be due to an unseen companion orbiting the donor star. The imprints left in the outflow carry information on the companion but also on the launching mechanism of these dust-driven winds. Aims. We study the morphology of the circumbinary envelope and identify the conditions of formation of a wind-captured disk around the companion. Long-term orbital changes induced by mass-loss and mass transfer to the secondary are also investigated. We pay particular attention to oxygen-rich i.e. slowly accelerating outflows, in order to look for systematic differences between the dynamics of the wind around carbon and oxygen-rich asymptotic giant branch (AGB) stars. Methods. We present a model based on a reduced number of dimensionless parameters to connect the wind morphology to the properties of the underlying binary system. Thanks to the high performance code MPI-AMRVAC, we run an extensive set of 70 threedimensional hydrodynamics simulations of a progressively accelerating wind propagating in the Roche potential formed by a massloosing evolved star in orbit with a main sequence companion. The highly adaptive mesh refinement we use enables us to resolve the flow structure both in the immediate vicinity of the secondary, where bow shocks, outflows and wind-captured disks form, and up to 40 orbital separations, where spiral arms, arcs and equatorial density enhancements develop. Results. When the companion is deeply engulfed in the wind, the lower terminal wind speeds and more progressive wind acceleration around oxygen-rich AGB stars make them more prone than carbon-rich AGB stars to display more disturbed outflows, a disk-like structure around the companion and a wind concentrated in the orbital plane. In these configurations, a large fraction of the wind is captured by the companion which leads to a significant shrinking of the orbit over the mass-loss timescale, if the donor star is at least a few times more massive than its companion. In the other cases, an increase of the orbital separation is to be expected, though at a rate lower than the mass-loss rate of the donor star. Provided the companion has a mass of at least a tenth of the mass of the donor star, it can compress the wind in the orbital plane up to large distances. Conclusions. The grid of models we compute covers a wide scope of configurations in function of the dust chemical content, the terminal wind speed relative to the orbital speed, the extension of the dust condensation region around the cool evolved star and the mass ratio. It provides a convenient frame of reference to interpret high-resolution maps of the outflows surrounding cool evolved stars.