  The decays of the $\beta$-delayed neutron emitters $^{137}$I and $^{95}$Rb
have been studied with the total absorption $\gamma$-ray spectroscopy
technique. The purity of the beams provided by the JYFLTRAP Penning trap at the
ion guide isotope separator on-line facility in Jyv\"askyl\"a allowed us to
carry out a campaign of isotopically pure measurements with the decay total
absorption $\gamma$-ray spectrometer, a segmented detector composed of eighteen
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 $\gamma$-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 $\beta$-intensity to states
above the neutron separation energy that de-excite by $\gamma$-rays, comparable
to the neutron emission probability. The competition between $\gamma$
de-excitation 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 $\beta$-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.
