My best guess from the description is that your melons suffered from squash vine borer. This insect bores holes into the stem of melons, squash and cucumbers and eats the inner lining. You can check for a flat, white caterpillar inside the stems on the infected plants. At this late stage you should pull up all the vines, roots and all if you can and place in a plastic garbage bag. Lay in the sun to heat up for a few days and kill the boring insects still inside the stems. Next year use floating row covers on top of the plants to keep the adult squash borer flies from landing on your plants and depositing the eggs that will hatch and bore into the stems.  Good fall cleanup and covering the crop is your best preventative measure. Good luck and Keep Growing, Marianne Binetti