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Asteroid binaries found amongst the Near-Earth objects are believed to have formed from rotational fission. In this paper, we aim to study the dynamical evolution of asteroid systems the moment after fission. The initial condition is modelled as a contact binary, similar to that of Boldrin et al. (2016). Both bodies are modelled as ellipsoids, and the secondary is given an initial rotation angle about its body-fixed $y$-axis. Moreover, we consider six different cases, three where the density of the secondary varies, and three where we vary its shape. The simulations consider 45 different initial tilt angles of the secondary, each with 37 different mass ratios. We start the dynamical simulations at the moment the contact binary reaches a spin fission limit, and our model ensures that the closest distance between the surfaces of the two bodies is always kept at 1 cm. The forces, torques and gravitational potential between the two bodies are modelled using a newly developed surface integration scheme, giving exact results for two ellipsoids. We find that more than 80% of the simulations end with the two bodies impacting, and collisions between the bodies are more common when the density of the secondary is lower, or when it becomes more elongated. When comparing with data on asteroid pairs from Pravec et al. (2019) we find that variations in density and shape of the secondary can account for some of the spread seen in the rotation period for observed pairs. Furthermore, the secondary may also reach a spin limit for surface disruption, creating a ternary/multiple system. We find that secondary fission typically occurs within the first five hours after the contact binary separates, and is more common when the secondary is less dense or more elongated.