We consider the dressing of exciton properties by strongly correlated electrons in gate-controlled twisted homobilayer heterostructures. The combined effects of Moire potential and Coulomb interaction support the formation of different strongly correlated phases depending on the filling. This includes charge-ordered metals and integer-population incompressible insulators. The coupling between the exciton and the electron splits the exciton resonance into attractive and repulsive polaron peaks. Analysis of the excitonic polaron properties over the different phases of the system reveals a discontinuous evolution of the spectrum with the formation of a double-peaked structure at the repulsive polaron branch. The double-peaked structure appears with non-integer fillings and is controlled by the energy separation between the quasiparticle states near the Fermi level and the high-energy doubloon excitations. Our results show that exciton polarons have distinct features of electron correlation and thus provide a direct characterization of the formation of correlation-driven insulators in gate-controlled heterostructures.