Let \(G/P\) be a complex cominuscule flag manifold of type \(E_6,E_7\). We prove that each characteristic cycle of the intersection homology (IH) complex of a Schubert variety in \(G/P\) is irreducible. The proof utilizes an earlier algorithm by the same authors which calculates local Euler obstructions, then proceeds by direct computer calculation using Sage. This completes to the exceptional Lie types the characterization of irreducibility of IH sheaves of Schubert varieties in cominuscule \(G/P\) obtained by Boe and Fu. As a by-product, we also obtain that the Mather classes, and the Chern-Schwartz-MacPherson classes of Schubert cells in cominuscule \(G/P\) of type \(E_6,E_7\), are strongly positive.