Cam, since your e-mail address is hidden, I can not contact you but you can. I would like to take a look at the church baptismal record that you have to take a stab at the JA?BROCKI surname on the document.
No idea why but sometimes the ancestors that were from historical Hungary wrote down Austria on their US documents. I have left you a post on Austria's board with map of A-H and Galicia.
Roman Catholic church baptismal record will be in Latin and that is what you have in the first name Joannes - Jan, John, Johann, Janos.
Z^ofia/Z^ofja equals to Sophia in English.
Sometimes a draft registration form would have a place of birth but that might be a dead end. The best thing is to find Zofia's arrival manifest. Have you followed up on SADOWSKI ancestor's record? Maybe he was from Zofia's or John Andrew's village or vicinity.
Surname GACEK shows up in Slovakia but not in the area of where the border of old Hungary and Galicia touch.
Did John Andrew have any siblings that came before or after him to America that you could follow their paper trail?Who was the family that he was coming to first time to an unknown place to him? Most likely that person was from a same location as he.
You already took a look at these EI arrival manifests? Gaczek, ZofiaSzczarnicza, Austria219121914 Gaczek, ZofiaSlowe, Austria619011907 Gaczek, ZsofiaPolerjeka1618881904