Gunk that hardens when you don't use it? An airbrush shouldn't stop working just because it isn't used. It's a straw, for cryin' out loud! Clean your airbrush properly and it won't "harden"!
You should be able to order a replacement part from Badger, or the store you bought it from, but consider sending the whole airbrush back to Badger for a tuneup. Likely they will replace the part for free. They'll also clean out any remaining gunk.
Expecting a free part because of human error is a bit much. Expecting a free part because it broke is quite another thing. Rick only mentioned that they will often do such a thing.