The fuel sloshes back & forth as you accelerate.
The fuel sending unit is sticking, or the wiring to the gauge is loose. These are the two most common causes of an erratic gauge.
a bad ground
Most vehicles have one fuse that protects all the gauges so if the fuel gauge is the only one not working, the fuse is probably okay. Most causes for a inop fuel gauge is a faulty fuel gauge sending unit in the fuel tank.
Fuel Pump
Defected fuel pump sending unit. The #1 cause is that the sulfur in todays gas coats the fuel leveling sensor and causes the fuel gauge to act up. It is a common problem that is starting to take place.
A bad o2 sensor can cause this problem in a car. A bad fuel pump, fuel filter or fuel line will also cause hesitation during acceleration.
When you turn a corner the guage fluctuates.
Try to find a fuel pressure gauge. The fuel pump sounds suspicious from your description.
The gas gauge measures the amount of fuel left in a vehicle's gas tank, not velocity, mass, or acceleration. It indicates how much fuel is remaining and helps drivers estimate how much longer they can drive before needing to refuel.
The normal causes of an inoperative fuel gauge is the sending unit in the tank. Since the truck is 15+years old you might as well replace the fuel pump when you have the fuel tank removed.
There are many things that could cause hesitation on acceleration in a vehicle. It could be the fuel pump, the gas pump, fuel pump relay, or an electrical problem.
There are numerous issues that can cause a lack of power during acceleration. Common causes include a clogged fuel filter, a clogged air filter, and dirty fuel injectors.