Context. The dynamical interactions between young binaries can perturb the material distribution of their circumstellar disks, and modify the planet formation process. In order to understand how planets form in multiple stellar systems, it is necessary to characterize both their binary orbit and their disks properties. Aims. In order to constrain the impact and nature of the binary interaction in the RW Aur system (bound or unbound), we analyzed the circumstellar material at 1.3 mm wavelengths, as observed at multiple epochs by the Atacama Large (sub-)millimeter Array (ALMA). Methods. We analyzed the disk properties through parametric visibility modeling, and we used this information to constrain the dust morphology and the binary orbital period. Results. We imaged the dust continuum emission of RW Aur with a resolution of 3 au, and we find that the radius enclosing 90% of the flux (R_90%) is 19 au and 14 au for RW Aur A and B, respectively. By modeling the relative distance of the disks at each epoch, we find a consistent trend of movement for the disk of RW Aur B moving away from the disk of RW Aur A at an approximate rate of 3 mas yr^‑1 (about 0.5 au yr^‑1 in sky-projected distance). By combining ALMA astrometry, historical astrometry, and the dynamical masses of each star, we constrain the RW Aur binary stars to be most likely in a high-eccentricity elliptical orbit with a clockwise prograde orientation relative to RW Aur A, although low-eccentricity hyperbolic orbits are not ruled out by the astrometry. Our analysis does not exclude the possibility of a disk collision during the last interaction, which occurred 295_‑74⁺²¹ yr ago relative to beginning of 2024. Evidence for the close interaction is found in a tentative warp of 6 deg in the inner 3 au of the disk of RW Aur A, in the brightness temperature of both disks, and in the morphology of the gas emission. A narrow ring that peaks at 6 au around RW Aur B is suggestive of captured material from the disk around RW Aur A.