Medical image super resolution (SR) plays a key role in improving diagnostic accuracy through the reconstruction of high-resolution fine anatomical details from low-resolution scans. Conventional CNN-based SR methods tend to lose edge fidelity and fail to effectively remove clinical artifacts. To address these issues, a new Transformer-Integrated Dual-Domain Residual Network (TDDRNet). It incorporates spatial and frequency domain information through dual-path residual blocks, augmented further through transformer-based attention mechanisms. The proposed Gradient-Aware Supervision (GAS) to push the network to refine edge transitions and high-frequency patterns leveraging gradient priors. Additionally, a novel optimizer, Curvature-Driven Lookahead Adaptive Moment Optimization (CLAMO) adaptively balances gradients leveraging curvature-aware feedback and Lookahead components to provide rapid convergence, enhanced generalization, and avoiding local minima. Extensive tests on the MRBrain dataset with a 4× scaling factor show the excellence of our approach, with a PSNR of 32.4 dB, SSIM of 0.956, and accuracy of classification of 98.6% with a loss of 0.08.