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The study of subtle effects on transport in semiconductors requires high-quality epitaxial structures with low defect density. Using hybrid molecular beam epitaxy (MBE), SrTiO$_3$ films with low-temperature mobility exceeding 42,000 cm$^2$V$^{-1}$s$^{-1}$ at low carrier density of 3 x 10$^{17}$ cm$^{-3}$ were achieved. A sudden and sharp decrease in residual resistivity accompanied by an enhancement in the superconducting transition temperature were observed across the second Lifshitz transition (LT) where the third band becomes occupied, revealing dominant intra-band scattering. These films further revealed an anomalous behavior in the Hall carrier density as a consequence of the antiferrodistortive (AFD) transition and the temperature-dependence of the Hall scattering factor. Using hybrid MBE growth, phenomenological modeling, temperature-dependent transport measurements, and scanning superconducting quantum interference device imaging, we provide critical insights into the important role of inter- vs intra-band scattering and of AFD domain walls on normal-state and superconducting properties of SrTiO$_3$.