Accurate segmentation of gastrointestinal (GI) organs in magnetic resonance enterography (MRE) is critical for diagnosing inflammatory bowel disease (IBD). However, anatomical variability, class imbalance, and low tissue contrast hinder reliable automation. This study proposes a dual-stage deep learning framework for organ-specific segmentation of GI structures from coronal MRE images to address these challenges. A publicly available MRE dataset of 3,195 coronal T2-weighted HASTE slices from 114 IBD patients was used. Initially, a DenseNet201-UNet++ model generated coarse masks for ROI extraction. A DenseNet121-SelfONN-UNet model was then trained on organ-specific patches. Extensive data augmentation, normalization, five-fold cross-validation, and class-specific weighting were applied to mitigate severe class imbalance, particularly for the appendix. The initial stage achieved strong organ localization but underperformed for the appendix; class weighting improved its DSC from 6.76% to 85.76%. The second-stage DenseNet121-SelfONN-UNet significantly enhanced segmentation across all GI structures, with notable DSC gains (cecum +23.62%, sigmoid +18.57%, rectum +17.99%, small intestine +16.06%). Overall, the framework achieved mDSC of 88.99%, mIoU of 84.76%, and mHD95 of 6.94 mm, outperforming all baselines. This framework demonstrates the effectiveness of a coarse-to-fine, organ-aware segmentation strategy for intestinal MRE. Despite higher computational cost, it shows strong potential for clinical translation and enables anatomically informed diagnostic tools in gastroenterology.