Engineering problem solving is central to real-world decision-making, requiring mathematical formulations that not only represent complex problems but also produce feasible solutions under data and physical constraints. Unlike mathematical problem solving, which operates on predefined formulations, engineering tasks demand open-ended analysis, feasibility-driven modeling, and iterative refinement. Although large language models (LLMs) have shown strong capabilities in reasoning and code generation, they often fail to ensure feasibility, which limits their applicability to engineering problem solving. To address this challenge, we propose EngiAgent, a multi-agent system with a fully connected coordinator that simulates expert workflows through specialized agents for problem analysis, modeling, verification, solving, and solution evaluation. The fully connected coordinator enables flexible feedback routing, overcoming the rigidity of prior pipeline-based reflection methods and ensuring feasibility at every stage of the process. This design not only improves robustness to diverse failure cases such as data extraction errors, constraint inconsistencies, and solver failures, but also enhances the overall quality of problem solving. Empirical results across four representative domains demonstrate that EngiAgent achieves substantial improvements in feasibility compared to prior approaches, establishing a new paradigm for feasibility-oriented engineering problem solving with LLMs. Our source code and data are available at https://github.com/AI4Engi/EngiAgent.