Yeah, Gore Tex is annoying because once it gets wet the stinkin stuff never dries here in the SE. I know cause most of my footwear has Gore Tex lining to my dismay (I buy footwear solely based on fit, and most of the stuff that fits me just happens to have Gore Tex in it).

At any rate, I have two suggestions that may help. For hiking in rain or after rain I use OR Rocky Mountain Low gaiters. These keep most of the water from going into my boots. Usually my boots will only get wet in a steady rain if I am actually hiking in it due to water running down my legs. Even the gaiters can't stop this from happening, but they do at least minimize the "damage". The other thing I have recently started doing due to issues I have from my feet getting wet is to rotate socks throughout the day where I wear one pair, and lash the other pair (or pairs) to the outside of my pack to dry. I then change into a dry pair during a break.