Try getting a cap or plug for the side that works and see if the other side moves. If you can, get someone else to push the pedal so that you can watch and see if it moves and if it moves smoothly.
Basically, I had that. I set the parking brake so the side that works (Passenger side) wouldn't move, then I stepped on the brake pedal and the drivers side didn't move. New caliper is on it and it works fine. I opened up the old one and saw nothing out of the ordinary, but I have never opened up a caliper with a parking brake integrated, so I may have missed something obvious. I have already turned in the core and I didn't take pictures.
I got the reman caliper from Ford because the price was just a couple dollars more than NAPA and I have an inherent distrust of autoparts store reman parts.
The book said that to bleed it, you mount it with just the piston side pad on and a rag on the other side and you push the brake, then you have to turn the piston back in. I just mounted it, hooked up the line and let it gravity bleed. Once I had fluid out the bleeder, I then bled it with my bleeder hose.
My bleeder hose is an 8 inch piece of clear plastic tube that I have tied in a loop. As long as the loop is up, you can slowly push and release the brake pedal and the loop holds the fluid so it doesn't suck the air back in. Well, I noticed that before I changed calipers, I could only give the pedal a short stroke, otherwise, it would suck all of the fluid back through my hose and I would get air. So after the new caliper was on, I could give it full strokes and it would only suck the fluid back a little bit.
I don't know what the problem was inside the caliper, but it was night and day difference.