Val,
The thought occurs to that Fanny's original surname was probably Shneyder (cf. Scheider = tailor in German). Shneyder is how the name is transliterated from Yiddish into English.
Zagare was in the Šiauliai district (and rabbinate) of Kaunas gubernia. As far as I can make out, nonew of the BMD records for that district were microfilmed.
I have justed looked through the 2,000 odd voter's records for Zagare - nothing there.
The problem is that our Jewish ancestors moved around a lot. A good relevant example is the family of the famous singer, Al Jolson. Born as Eisa Yaelson in Serdzius (west of Kovno) in 1886, he was son of parents who had married outside Seredzius, and had both come from separate towns even further afield.
Did you ever see the episode of WDYTYA on David Suchet? Similar story.
Justin