I'm guessing that the person was joking. Chicago isn't constructivist at all. You have the realists (Pape, Mearsheimer, etc.) and liberals (Snidal, Lipson, etc.). They did NOT produce Wendt, so every IR theory course makes you read Social Theory of International Relations so that they can rip it in half, but other than that there ain't much constructivism going on.
Hondo - You should have a very reasonably chance at all of the schools that you listed. Your verbal GRE will hurt you some, but I don't believe that any of those schools listed have a strict cutoff (filtering out anyone below a certain score). Best of luck.
If you have problems getting in this year, many schools have 'cash cow' MA programs that let you go and prove your worth at an elite school. Just an idea. I took this approach because I too went to a 4th tier state school. Letters from one of these schools will go a long way.