We study the properties of one-dimensional exciton systems in which the commonly made Heitler-London approximation (HLA) is relaxed. The nonresonant interaction terms which then exist, mix the multi-exciton bands of the HLA. Our approach is based on the exact diagonalization of the Hamiltonian, which is possible using the Jordan-Wigner and Bogoliubov transformations. Exact expressions for transition dipoles between multi-particle states are given. Results of our exact theory for the ground state and one-particle energies, the superradiant enhancement, the pump-probe spectrum, and the linear absorption to multi-particle states are compared quantitatively to the HLA, to the Bose approximation (where the excitons are treated as bosons), and to perturbation theory. In this comparative study, we use parameter values that are relevant to much studied quasi-one-dimensional J aggregates, such as PIC and TDBC. We find that for these systems the strongest effects of the HLA occur in the oscillator strengths of the various optical transitions. In particular, the exciton delocalization length derived from the experimentally observed superradiant enhancement is overestimated by roughly 10% due to the HLA. Also, the transition between the ground state and three-particle states, which is strictly forbidden in the HLA, does obtain a finite oscillator strength due to the non-resonant interactions.