This is almost certainly because of the limited slip rear differential. I've owned two Expeditions, a 99 and an 04. both of them had the problem. You can most likely fix it just by adding the friction modifier to the differential. You can get it at your Ford dealership for about $4 per bottle. It smells godawful, and the smell seeps through the bottle, so don't leave it sitting in your car.
I'm not sure about the '97, but on the '99 there is only one plug on the differential. It's on the left side. You remove it by simply sticking a 3/8ths drive ratchet into the square hole and unscrewing it. Don't lose the little brass gasket thingy. Then cut off the tip of the bottle and squirt it into the differential. A little fluid might drain out when you pull out the plug. Don't worry about it. The fluid is only supposed to be filled up to the bottom of the plug hole.
On my 2004 Expedition it actually took two bottles to get the noise to go away. However, I bought it for an absolute steel. I only paid $8500 for an 04 because the owner thought it had serious rear end problems. I paid his asking price. I didn't talk him down. But two bottles of friction modifier later, and it drives like brand-new.
After you add the friction modifier it will take a couple of hundred miles of driving before it's back to normal.