Context. High-resolution observations reveal that the outflows of evolved low- and intermediate-mass stars harbour complex morphological structures that are linked to the presence of one or multiple companions. Hydrodynamical simulations provide a way to study the impact of a companion on the shaping of the asymptotic giant branch (AGB) star out-flow. Aims. Using smoothed particle hydrodynamic (SPH) simulations of an AGB star undergoing mass loss, which also has a binary companion, we study the impact of including H I atomic line cooling on the flow morphology. We also study how this affects the properties of the accretion disks that form around the companion. Methods. We used the PHANTOM code to perform high-resolution 3D SPH simulations of the interaction of a solar-mass companion with the outflow of an AGB star, using different wind velocities and eccentricities. We compared the model properties, computed with and without the inclusion of H I cooling. Results. The inclusion of H I cooling produces a sizeable decrease in the temperature, up to one order of magnitude, in the region closely surrounding the companion star. As a consequence, the morphological irregularities and relatively energetic (bipolar) outflows that were obtained without H I cooling no longer appear. In the case of an eccentric orbit and a low wind velocity, these morphologies are still highly asymmetric, but the same structures recur at every orbital period, making the morphology more regular. Flared accretion disks, with a (sub-)Keplerian velocity profile, are found to form around the companion in all our models with H I cooling, provided the accretion radius is small enough. The disks have radial sizes ranging from about 0.4 to 0.9 au and masses around 10−7−10−8 M⊙. For the considered wind velocities, mass accretion onto the companion is up to a factor of 2 higher than predicted by the standard Bondi Hoyle Littleton rate, ranging between ~4 to 21% of the AGB wind mass loss rate. The lower the wind velocity at the location of the companion, the larger and the more massive the disk and the higher the mass accretion efficiency. In eccentric systems, the disk size, disk mass, and mass accretion efficiency vary, depending on the orbital phase. Conclusions. H I cooling is an essential ingredient to properly model the medium around the companion where density-enhanced wind structures form and it favours the formation of an accretion disk.