In the open ocean, the formation of large waves is non-linear. This non-linearity modifies the shape and structure of steep waves relative to that expected in a linear model.Work by Adcock et al. ["Nonlinear dynamics of wave-groups in random seas: Unexpected walls of water in the open ocean," Proc. R. Soc. A 471(2184), 20150660 (2015)] used a numerical model to show that, for a wave spectrum representative of that of a storm in the North Sea, non-linear dynamics resulted in the following changes to the shape of extreme wave groups relative to a linear model: expansion of the wave-crest laterally, contraction of the wave-group in the mean wave direction, and a movement of the largest wave to the front of the wave-group. They found only moderate elevation above that expected by the linear model. This paper extends this work to explore the influence of spectral bandwidth and directional spreading on these results. We find that sea-states with low directional spreads are more likely to see significant non-linear changes to extreme wave-groups. However, rather surprisingly, extreme waves formed in seas with broader spectra generally, although not always, see greater non-linear changes for the range of spectra studied here. We also observe that the lateral expansion to the wave-group occurs even for relatively modest sea-states and is a predictable feature. By contrast, the contraction of the wave-group in the mean wave direction only occurs occasionally in the steepest seastates- although when this is triggered the change can be very dramatic.