Mixed-Integer Linear Programming (MILP) is a key technology for solving a wide range of real-world combinatorial optimization problems. However, existing machine learning methods addressing these problems primarily focus on node features while neglecting edge features, limiting their ability to extract complete constraint information. To address this limitation, this paper proposes a novel solution framework named SHARP based on an edge-enhanced Graph Neural Network accelerated by the Sinkhorn algorithm. This framework effectively fuses node and edge representations by incorporating edge features into the attention mechanism to better learn the underlying patterns of MILP. Furthermore, to overcome the shortcomings of traditional methods in problem scale generalization and hyperparameter tuning, an adaptive Regret-Greedy algorithm is designed to enhance solution feasibility and quality by dynamically adjusting variable assignment strategies. Experimental results on combinatorial auction and item placement datasets show that the proposed framework achieves performance improvements of 24.88% and 5.86% on the Primal Integral metric compared to the Gurobi and SCIP solvers, respectively, and a 17.19% improvement over the current state-of-the-art ML method.