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Did you replace the heater core the same way, i.e via cutting into the box from the glove compartment? If yes, did you have to remove the air bag?Likely, you have a 1999 with "rear A/C". So, you have an auxiliary condenser in the rear. (Driver's side rear sidepanel, just behind the rear wheel.) But this means that the A/C compressor is working (in the engine compartment), and the rear condenser and electric fan motor in the rear are working. In the front, I assume that the fans are OK since your said the front "blows" hot. Sounds like your A/C blend door has broken, and is stuck in the "Hot" position. This is common in 1st Gen Expeditions. (The OEM dealer/official repair is quite extensive (and expensive) because you have to pull the dash to replace the A/C blend door. (The A/C blend door is a "channel vent" control in the "Air / Heat box" under the dash, on the passenger side, generally behind the glove box.
Because the rear, and the front are on the same pressurized refrigerant circuit, and the rear is working, the "front" condenser is also probably good. (If it was leaking, you would see green dripping stuff either inside, on the floor passenger side where your feet go, under the dash, or you run the A/C, and look under the car where the vehicle usually drips water (behind the front passenger side wheel, at the fire wall, and see if it is dripping green. But too me, it sounds like the front A/C is "also" OK. The reason you blow "hot" up front, is that the vent control door in the heat box is not moving to the "cold" position when you turn the hot/cold knob to cold (or set the cold temp if you have electronic air control.)
The blend doors on pretty much 100% of 1st Gen's fail by this time. It is made of plastic, and it was not a good design. The part is only $25-75 dollars. All the other problem is the labor. (You do not have to discharge your system, or anything else.)
I have done the repair the "factory" way, and it was a lot of work. There are a few videos on youtube on how to change that door out in a "cheater" way. In the cheater way, you remove the glove box, and then cut into your heat/cold box on the left/back as seen from inside passenger seat (e.g. center of vehicle side, rear side) on the upper part of the black box clamshell, so that you can remove the blend door, replace it, then put the clamshell back into place, and then "tape" it back on using aluminum duct tape. (There is a fair amount of air pressure there because the air blower inserts into that box as well (at the other end.)
Before ripping the box apart, I would also check your "blend door" actuator. This is the electronic component (the vacuum or electric motor driven) actuator that causes the door to turn, and make sure its working. That part is located under the dash, and under the center console on a 1999 Ford Expedition. If you take the two plastic kick panels out from the drivers and passenger side, you can see it. Its a little black servo box. That is the other item that can fail that also controls the same process. One of them the servo/motor fails, the other the door connection "mate" (to the servo) cracks/breaks. Either results in the door not moving.
1999 FORD EXPEDITION 5.4L V8 Heater Air Door | RockAuto
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At this URL, you can see the door. The portion that breaks, on the replacements, is made or coated with aluminum, (so that it won't break again), at the base of the shaft. The "cross" that looks sort-of like a Philips screw head (is where the cross shaped top of the actuator fits into it.) That cross on the original is made of 100% plastic, and the shaft cylinder breaks. The actuator turns, but the door does not. If the actuator breaks, then the actuator does not turn, so then the door doesn't move.
To replace the actuator, you are supposed to remove the center console, driver's seat, and kick panels, and then undue the screws that hold the actuator in place. You can do it just removing the kick panels by using some elbow grease, but some of the screws are hard to get at.
Get some quotes if you are having it done, big variance in price I would imagine. I have done this job twice essentially, because I had to replace my heater core (in the same box). Wish I had replaced the blend door at the same time, but I didn't, and the actuator, which broke later (but which is a different job.)