The formation and evolution of ring structures in galaxies are crucial for understanding the nature and distribution of dark matter, galactic interactions, and the internal secular evolution of galaxies. However, the limited number of existing ring galaxy catalogs has constrained deeper exploration in this field. To address this gap, we introduce a two-stage binary classification model based on the Swin Transformer architecture to identify ring galaxies from the DESI Legacy Imaging Surveys. This model first selects potential candidates and then refines them in a second stage to improve classification accuracy. During model training, we investigated the impact of imbalanced data sets on the performance of the two-stage model. We experimented with various model combinations applied to the data sets of the DESI Legacy Imaging Surveys DR9, processing a total of 573,668 images with redshifts ranging from z_spec=0.01-0.20 and mag_r_<17.5. After applying the two-stage filtering and conducting visual inspections, the overall precision of the models exceeded 64.87%, successfully identifying a total of 8052 newly discovered ring galaxies. With our catalog, the forthcoming spectroscopic data from DESI will facilitate a more comprehensive investigation into the formation and evolution of ring galaxies.