Aaah, the "weeper" - nice one. While the use of the term is new to me, the phenonemon is not. I have occasionally experienced this right after a shave, albeit not to the degree that the OP describes, on certain sensitive "thinner-skinned" areas, like below the nose and on the cheeks. To the best of my knowledge, it only happens when I'm:
a) In a real hurry;
b) Using a slightly less than optimally sharp blade;
and, get this:
c) When, due to a) I use a Gillette Fusion razor to finish off difficult spots!
I surmise that it is indeed very likely I apply too much pressure throughout this rushed procedure.