We establish a general version of the strong KPZ universality conjecture near the axis for random walks in a random environment (RWRE) on $\mathbb{Z}^2$. For an i.i.d. elliptic random environment, we consider the quenched large deviations probabilities for trajectories starting at the origin and arriving at time $n+[n^a]$ to the position $(n,[n^a])$ and show that, if the logarithm of the right-jump probability has a finite moment of order $p>2$, then for $a < \frac{3}{7}(1-\frac{2}{p})$ the fluctuations of these propabilities are asymptotically governed by the GUE Tracy-Widom distribution. Our results are based on a comparison between RWRE and a last passage percolation model, whose asymptotic fluctuations near the axis were previously established independently by Bodineau-Martin and Baik-Suidan. Furthermore, we obtain also the full convergence to the directed landscape in this regime based on the extension of the aforementioned results to this setting by McKeown and Zhang.