The β-delayed neutron emission of ^{83,84}Ga isotopes was studied using the neutron time-of-flight technique. The measured neutron energy spectra showed emission from states at excitation energies high above the neutron separation energy and previously not observed in the β decay of midmass nuclei. The large decay strength deduced from the observed intense neutron emission is a signature of Gamow-Teller transformation. This observation was interpreted as evidence for allowed β decay to ^{78}Ni core-excited states in ^{83,84}Ge favored by shell effects. We developed shell model calculations in the proton fpg_{9/2} and neutron extended fpg_{9/2}+d_{5/2} valence space using realistic interactions that were used to understand measured β-decay lifetimes. We conclude that enhanced, concentrated β-decay strength for neutron-unbound states may be common for very neutron-rich nuclei. This leads to intense β-delayed high-energy neutron and strong multineutron emission probabilities that in turn affect astrophysical nucleosynthesis models.