Dear Fellow,
You can't move $E$ (!), hence there is no contradiction with it having self-intersection -1. Indeed, if you take a normal vector field along $E$, it will necessarily have degree -1 (i.e. the tolal number of poles is one more than the total number of zeroes), or (equivalently), the normal bundle to $E$ in the blown-up surface is $\mathcal O(-1)$.