Welcome aboard Bob.
If I understand you correctly: When you open the driver's door on the mounty the chime sounds even if the lights aren't on and the key is not in the ignition. If you have the key in the ignition and open the door the chime sounds, and if you remove the key, the chime continues to sound. Is this correct?
3 Things come to mind: headlight switch, ignition switch, broken wire(s) in the boot.
Since the headlights being on, or the key being in the ignition will cause the chime to sound, one of these switches may have a loose connection or be faulty.
I had some electrical issues with my 02 xlt. The dome lights would stay on until the battery saver feature turned them off, the driver side puddle light didn't work, the keypad on the door would set off the alarm when I pressed it, the door ajar light would flash and beep at highway speeds, the radio would shut off as soon as I turned the key off, etc.
A lot of electrical issues can be traced back to broken/damaged wire(s) in the boot between the body and the driver's door. I found a broken wire and repaired it and all is back to normal!
Here is the thread for removing the door panel:
http://www.explorerforum.com/forums//showthread.php?t=303524
Here is a great thread for checking for broken/damaged wires, thanks to
synyster for this awesome write-up:
http://www.explorerforum.com/forums//showthread.php?t=270081
I'm just shooting in the dark here.

Hopefully someone with more knowledge than me will check in. Good luck with the fix and be sure to keep us posted.