Hunt down the "beeper module" or chime and yank it out. My girlfriend's t-bird had a door ajar sensor that was on the brink and the chime would be going nonstop. I just found the module (I think...it's on the pass side near the cabin blower but you could always follow the beep). Sounds like you have a short. I would disable it just in case it decides to trigger for no reason (unplug connector on back of airbag).
Actually, it is supposed to do this from the factory. If you are having problems with the air bag, a idiot light will come on. If that light fails, the beep will kick in. I don't want to trace down the beep because it is coming from the center of the dash above the radio. It would probably be a pain to get to, so I would rather just fix the problem because the sensors are easier to get to. I just don't know how to fix it.
The center dash isnt all that hard to get to. I see 1989 and 1991 in your sigs. Same generation as my 91. I just finished pulling the _entire_ interior out of my 1991 to tear up the floorboard soundproofing. I would pull out the radio and look up if it's coming from that area. Radio is just 2 easy clips. Should be easy enough to get to.
But seriously:
If the airbag light is on...chances are that system is no longer working and wouldnt fire. I doubt you would want to go in and try to debug that particular system due it is rather explosive nature if you happen to short the wrong thing (though I would think it take more than 12V to fire an airbag).
I intentionally removed my airbag system a long time ago and it's quite a bit of wiring and at least 2 baby blue modules under the dash and a connectors and some harnesses going along the steering column. Lots of wiring. I didnt seem to have any impact sensors on the bumper so I'm not sure how the fords sense impact but this car _was_ rebuilt from a collision at some point.
if i remember correctly, it's a blue box located behind the a/c controls in the center console - as you noted. It wasn't the easiest thing to disconnect given the location, but I managed to get it out w/o breaking anything. I think you can reach up from behind a little, but I can't remember. If you can, it might just be better to fix it. I'm not positive what mine was for now, but I think it was the airbag.
Thank you. That was informative. Once I put the transmission back in I will use that to diagnose the beep. In the manual it says it will beep 5 times. That is what mine does.