Why is the current draw dropping in water electrolysis
If you look at the copper wires after the electrolysis, the cathode (connected to the negative pole of the current source) will be white. This electrode is coated with Mg(OH)2.

The low activity of copper on the electrochemical scale makes it unlikely (impossible) for copper to displace magnesium from an aqueous salt.

However, the initial voltage is high enough to theoretically evolve metallic magnesium, which would instantly react with water to produce H2 and Mg(OH)2, which will precipitate because of its low solubility. The Mg(OH)2 adheres, at least to some degree, on the copper electrode and interferes with current flow. The resistance to the electrolysis at the same voltage leads to a lower resulting current.

Even at electrode potentials below that needed to evolve metallic Mg, current carried by the aquo Mg++ ion will result in an increased pH and deposition of Mg(OH)2 around the cathode. An interesting question is whether the deposit from a low voltage electrolysis has the same adhesion as from a high voltage experiment. Maybe low voltage electrolysis would deposit Mg(OH)2 farther out in the solution and not be so blocked off by adhesion to the electrode.