You would have a civil war on your hands if you tried to drain the Great Lakes to cool down Texas. Or irrigate the Great Plains, for that matter. Various proposals have been made over the years for large scale water diversions from the Great Lakes, with the result that the Great Lakes states have formed a compact against any such attempt.
Doesn’t it flow over Niagra Falls into the Atlantic Ocean anyway? How is the Great Lakes “drained” exactly? I seriously doubt that piping water into these reservoirs would drastically reduce the amount of water there.