The decays of the β-delayed neutron emitters 137I and 95Rb have been studied with the total absorption γ-ray spectroscopy technique. The purity of the beams provided by the JYFLTRAP Penning trap at the ion guide isotope separator on-line facility in Jyväskylä allowed us to carry out a campaign of isotopically pure measurements with the decay total absorption γ-ray spectrometer, a segmented detector composed of 18 NaI(Tl) modules. The contamination coming from the interaction of neutrons with the spectrometer has been carefully studied, and we have tested the use of time differences between prompt γ rays and delayed neutron interactions to eliminate this source of contamination. Due to the sensitivity of our spectrometer, we have found a significant amount of β intensity to states above the neutron separation energy that deexcite by γ rays, comparable to the neutron emission probability. The competition between γ deexcitation and neutron emission has been compared with Hauser-Feshbach calculations, and it can be understood as a nuclear structure effect. In addition, we have studied the impact of the β-intensity distributions determined in this work on reactor decay heat and reactor antineutrino spectrum summation calculations. The robustness of our results is demonstrated by a thorough study of uncertainties and with the reproduction of the spectra of the individual modules and the module-multiplicity gated spectra. This work represents the state-of-the-art of our analysis methodology for segmented total absorption spectrometers.