try this

1.start the vehicle and let it idle
2.fill the reservoir to the full mark,leave the cap off of the pump
3.from outside the vehicle turn off the engine and, with a flashlight,look inside of the reservoir.
You are looking for the fluid level to fluctuate or even spill out of the reservoir.Also look for a cloudy fluid.This usually takes about 15 seconds after you turn the engine off and it almost looks like the fluid is carbonated.
If any of these things happen you have air in the system.95% of all power steering noise is caused by air.One of the easiest ways to purge air is to

1.start the vehicle and turn the steering wheel to the full left posision.
2."Bounce" the steering wheel off of the steering stop a few times
3.emediately after turn off the engine and let the vehicle sit for 15min(Never turn the steering wheel with the engine off because all you are doing is pumping more air into the system)
4.start the vehicle and do the same process again but on the full right posision.
5.alternate between the 2 directions several times .

There is a very good chance this will purge the air.It works on 95% of the cars I work on.

One more thing to consider.If the noise is there even when you dont turn the steering wheel,It could be a bearing on one of your pulleys.This sounds very similar to power steering noise.Use a stethescope or a long screwdriver to pin point this.A screw driver works almost as good as a stethescope if you put your ear to the end of the handle and put the blade near the mounting location of the pulley in question.

I hope this helps and it probably wont take you half as long to figure out the noise as it did to type this.I suck at typing.

 

Id mark it and pay close attention to it position starting with the wheels straight but it may be keyed so that it will only go on one way. As to air still in there yes Its possible. Air can create a blockage that fluids cant penetrate or pass by.

 

I got under the pickup today to start taking off the pitman arm to test, and in doing so, I finally saw some fluid leaking from the steering gear box. Not a lot, just enough to make about 1 drop every 2 days if I had to guess. Does this kind of point to the gear box being the problem?

 

Not necessarily mine leaks worse than that but works fine. Bad thing is you can't replace that seal so far as I know. I will get one from a salvage yard when I get ready to fix it. You may want to do that as well if you determine its binding up and causing your problem.

 

Well, a friend and I jacked the front end of the pickup off the ground, leaving both wheels free to turn. It didn't bind at all. So I took it to a tire store to see if they could see anything, and they said pretty much all my ball joints are bad and need to be replaced. Do you think this is something that could cause the binding when turning? The tires are fighting against each other?

 

Absolutely and if there worn out you need to fix it even if its not the problem. Hope you get to the bottom of it.

 

I'm happy to report that after replacing all the ball joints on the pickup, the steering binding issue is gone! Thanks everyone for the suggestions!!

 

Way to go bud.