Low-background searches for astrophysical neutrino sources anywhere in the sky can be performed using cascade events induced by neutrinos of all flavors interacting in IceCube with energies as low as similar to 1 TeV. Previously we showed that, even with just two years of data, the resulting sensitivity to sources in the southern sky is competitive with IceCube and ANTARES analyses using muon tracks induced by charge current muon neutrino interactions-especially if the neutrino emission follows a soft energy spectrum or originates from an extended angular region. Here, we extend that work by adding five more years of data, significantly improving the cascade angular resolution, and including tests for point-like or diffuse Galactic emission to which this data set is particularly well suited. For many of the signal candidates considered, this analysis is the most sensitive of any experiment to date. No significant clustering was observed, and thus many of the resulting constraints are the most stringent to date. In this paper we will describe the improvements introduced in this analysis and discuss our results in the context of other recent work in neutrino astronomy.