Thanks for the tip synful1! After some digging around, I noticed that some gun manufacturers just recommend "anything not containing silicone." I had some Marvel Mystery Oil laying around, and I checked the FAQ site for it
http://www.marvelmysteryoil.com/index.php/site/faq/, and it says that MMO does not contain silicone. So I tried a few drops on the gun I have, which freed it up immediately, and seemed to spray just fine on some test panels I'm practicing on. I'm no expert, and the paint didn't quite come out like I had hoped, but I don't think the MMO had any bad effects, so I may buy a precision oiler at Harbor Freight, and put a drop or two of oil on both the main needle and the air valve needle each time I spray. I already use MMO on all of my air tools, so its nice to know I can stick with it as a product for my spray guns as well.