What do hot/warm wires mean?
If the wires are warm throughout then the gauge is too small. But if the wire is too hot close to the unit that tells me the unit is too hot and the wire is heated by conduction of the heat not from the copper losses.

I suspect your problem is lack of cooling on the hot side . These devices only generate a temperature difference between the hot and cold side so if you don’t remove the heat with the fan and proper heat sink on the hot side then the cold side will not reach expected  temperatures. Please measure with this thermometer both the hot and cold side of the device and not just the air temperature. You also need a conduction plates for the cold side in order to spread that heat absorption or cooling effect. Please report your test results. Using a CPU heatsink and fan and use AWG16 wire. The closer you can get the hot side to room temp, the colder the inside will get. Even try water cooling plumbing or use mineral oil instead of water with a tiny pump to a reservoir 

or a like they do in Mercedes SUV options make a car beverage cooler and warmer for cold and warm beverages on each side with a copper or alum sheet cylinder on either side !!  Probably a $500 option hah