Jimard, are you not making drain holes in the bottom half of your jugs? That will keep your seeds from being drowned. It won't matter how much rain, snow, or frost gets in the jugs then. I use my husband's soldering iron to poke holes. I tried heating a nail but that took too long. You could use the point of a knife to make holes or slits.
Last year I used the egg carton cardboard. Big mistake. Too soggy and fell apart. I would think poking holes in the Styrofoam ones would be better. Just a thought.