I have done a couple where the cooler is pissing coolant out into the exhaust, but the valve and intake have been dry.
Fill the cooling system, jack the ass end of the truck WAY up in the air, pull the EGR valve out, and pressurize the cooling system.
Look down in there and see if there's water pooling in the intake. Perhaps it's leaking at the back of the cooler, which is causing it to just run into the exhaust as opposed to the intake.
You could also try separating the exhaust at the manifold on the RH side, not much, just enough to allow liquid out, and try and determine which way it's flowing. I know that's not the most practical of ideas, but I'm also in the camp of you've got a blown cooler.
Also the oil cooler (or heat exchanger) and the screen underneath (6a642 and 6c683) should always be replaced during the course of this repair.