It can also mean that you are angry (rather like 'spitting blood'.) Apparently when someone is angry they spit foam, which may look like feathers. I also found this for the 'thirsty' definition:
Where it came from is suggested by Charles Wilson, whose childhood was spent in the southern USA: �I have often heard the expression, �I�m so thirsty I feel like I have a mouthful of feathers�, which strikes me as wonderfully descriptive�. Alice Battles and Joel Showalter mentioned �spitting cotton�, another way of expressing dryness once common in the American south, which is clearly a related idea.