Welcome to BYC. You are to be congratulated on having an 11 year old hen in such good condition until now. It's very hard to say what exactly is the matter, but it does sound a bit neurological. Drooping wings can be a sign of general, weakness, but the staggering sounds worrisome. I might suggest that you try some vitamins in the water, and give her a little egg each day for some extra protein. A complete16% protein chicken feed would be good for her. Worming her and checking for mites would also be helpful. She may be in her final stages of life, and I would try to keep her comfortable.
X2