Over on the "serious Explorations" forum, there are many threads about teh door switch wire bundle chafing in 2000-2007 Explorers and Aviators. Pulling the rubber boot back and a flashlight may let you see the bundle.

The door open/closed trickle test is most often used to see this.

Keep us in the loop. I'm on my 3rd battery in 2 years myself.......