You mentioned a white plastic piece sitting over the heater core, that you had to move out of the way. Any chance that it looked like the plastic piece on the left side of this photo? That device is connected to the temperature setting control on the dashboard / instrument panel, and controls the location of the blend door. It's possible you've managed to get it disconnected somehow. When I replaced the heater core last weekend, as seen in the picture, I had to set the temperature setting all the way hot (or cold, I don't remember) so that the white plastic device was fully out of the way. Otherwise, the heater core couldn't be removed because the plastic piece was in the way.