Hi! I had a 2001 Expedition and it did *exactly* what you describe. I tried to get it disabled so it would just quietly lock when pressing the button once and only chirp if I pressed it again. This way I could avoid waking the neighborhood just to lock the vehicle.
The dealer told me it was designed to chirp when you pressed the button, even just once, and there was nothing to do about it.
So, that is the way we left it. I always thought that Ford should have made it a user configurable item and still find it hard to believe there was not another solution. But, I never pursued it. Now the '01 is gone to car heaven and is being recycled...

....
P.S. The 1999 Expedition I owned and the 2004 both work as I like: One button press and the vehicle locks. Press it twice and it locks and chirps once.