All the ones I have seen are on all the time the headlights or park lights are on. They should not go off at any time unless you turn the other lights off.
I suspect a wiring problem at a trailer plug. Seeing this is a dually, I am sure you probably have trailer plugs, and may even have one in the bed for a goose neck or 5th wheel. Some people's wiring practices are not very good if these where added in later.
If the ground for these little lights happens to be hooked to the reverse lights, they would go out when the reverse lights are activated.
1989 F250 XLT Diesel