The separation of overlapping objects presents a significant challenge in scientific imaging. While deep learning segmentation-regression algorithms can predict pixel-wise intensities, they typically treat all regions equally rather than prioritizing overlap regions where attribution is most ambiguous. Recent advances in instance segmentation show that weighting regions of pixel overlap in training can improve segmentation boundary predictions in regions of overlap, but this idea has not yet been extended to segmentation regression. We address this with Overlap-Aware Segmentation of ImageS (OASIS): a new segmentation-regression framework with a weighted loss function designed to prioritize regions of object-overlap during training, enabling extraction of pixel intensities and topological features from heavily obscured objects. We demonstrate OASIS in the context of the MIGDAL experiment, which aims to directly image the Migdal effect--a rare process where electron emission is induced by nuclear scattering--in a low-pressure optical time projection chamber. This setting poses an extreme test case, as the target for reconstruction is a faint electron recoil track which is often heavily-buried within the order(s)-of-magnitude brighter nuclear recoil track. Compared to unweighted segmentation regression, we demonstrate OASIS's novel overlap region-targeted loss function weight to be the single most important training weight for improving intensity and topological reconstructions of the low-energy electron tracks that tend to be most dominated by pixel overlap. Averaging over eight training campaigns, we further show the addition of overlap-targeted weights to improve median intensity reconstruction errors from -41.1% to -13.3% for these low-energy electrons. These performance gains demonstrate OASIS as a generalizable methodology for recovering obscured signals in overlap-dominated regions.