Wow, this must be very frightening, it's so obvious and unsightly. I thought of several things - you may have been developing these for a while but something about the Liver-Flush caused flushing of your face so that the spider veins are more noticable. Hot packs and running would bring more blood to the skin so they might look worse. You may not be adsorbing the Vit K so you may have a maladsorbtion problem. When my sprue gets bad I bruise very easily probably because my Vit. K levels are low. Are bruising easily?
I have found that when I get cramps (not always painful) which make it more difficult for the intestinal contents to go through the digestive system and I massage out the blocked areas that when the cramps release my face and upper body flush. This would make spider veins more prominant and the purge would unblock your intestines.