How to clean a doll with a stuffed body and plastic head?

I have a Madam Alexander stuffed doll with a cloth body and a hollow plastic head. The head is softish plastic and held on with a zip tie thing.

She was stored in attic during a move and now she has water stains over much of her body (the clothing doesn't seem to be stained). Other than this she's in pretty good condition, considering all the use she endured. Does anyone know how to get the water stains out? Or at least get them out of her hand, the rest don't show through her clothes.

I can't just throw her in the wash because she has a sqeaker and those "sleepy" eyes that move, plus she's about 23 years old.

She's precious to me becuase I loved her dearly as a child and my grandmother gave her to me. Please help!

She's basically this doll (only an older version with simpler clothing) http://www.madamealexander.com/product/0/29200/_/H...