I think this has to be either the heater core is obstructed or the blend door is broken. I know you say you've replaced the heater core, but did YOU do it, or did a "professional" do it? And I know you (sort of) say that the hoses are hot on both sides, but maybe there's enough flow to make the hoses feel hot but not enough flow through the core to blow hot air into the cabin?
I'd check the blend door actuator first. Don't know if it's the same on your Taurus, but on my '97 there is a silver sheet-meal "bar" or sliver-strip under the dash just below the ashtray/radio area that the blend door actuator uses to raise the door up and down. When my actuator broke (plastic gear tooth cracked) it could no longer raise this sliver-bar for heat. For a while there (until I installed the new actuator), I was pushing the bar up manually (for the winter) and keeping it propped up with a small piece of plastic that I jammed into the sliver bar's slotted hole at the bottom. See if you can manually push this bar upward (propping the door up for heat). If you can, your actuator ($40) might be history. Turn the temp knob on the dash and watch the sliver-bar. Is it moving at all? With my car, it would move part of the way, then reach the broken gear tooth, and fall down (blocking heat flow).
Next, I'd flush the core. It might be "new", but it also might be obstructed. If the core is clean and the actuator is working, maybe you could have a problem with the vacuum system that controls the dash airflow. I've never had a problem with mine, but I'd think that if there WAS a problem with this, you'd at least get airflow SOMEWHERE - so I'd suspect this last.