I had a 2003 Supercrew F-150 5.4L/3.73 axle truck when we bought our 5500 pound dry TT. We towed for 2 seasons with it. You'll find that you have two acceleration gears and one cruising gear. It will slow down on hills/overpasses, so you will spend a fair amount of time in second gear. As long as you are not in a hurry, the truck will do fine at under 5k dry. Remember, it's a 17 year old truck. How well it was maintained is critical to how it will perform and how reliable it will be.
Flush all the fluids. If it is 2WD, consider changing the axle ratio to 4.10 which is what I thought put the engine in a better speed in 3rd at 65mph.