To those who were following this thread....
As I said, the alarm quit the other day and caused a message on the display to appear every time the vehicle was put into reverse. It read something like 'check reverse sensor alarm'.
I picked up a new "beeper" at our local Ford dealer (about 15 bucks) and replaced the old one. Everything is working perfectly now.
The beeper is attached directly to the backup module which is located behind the rear passenger seat. There is a flat panel between the seat backs and the bed of the truck. Simply remove the panel and insulation sheet and it will be located near the bottom on the passenger side. Very easy to do. All you'll need is a 7mm nut driver for the two screws located at the bottom inside of the storage compartment on the drivers side behind the rear seat. By the way, the beeper is fastened by two plastic clips and sticky tape.
I hope this information will help someone else with the same problem we had.