Perhaps someone else can chime in about the more recent records, but here is the info found in the 1875 census. This was a census that was created as a result of the previous year's law about implementation of a mandatory conscription of men into the imperial army. The census only consisted of eligible men and didn't have the entire population documented. For example, women were not included.
You probably know this, but just to reiterate, names like Herman, Irene, Evelyn, Simon, Gertrude are Americanized and not what your ancestors were called in Podolia.
Solomon might have been called Shlioma, Simon - Shimon, etc. The others you would have to find through the original naturalization or other documents.
It's possible that Duvid below is the same David you mention, but probably not possible to tell without having some additional documents. Also, not sure how Ios is related to Leybish.
The 1875 census for Novaya Ushitsya
Name - previous census - current age
Ios s.o. Duvid - 24 - 40
Duvid (son) - 1 - 19
Srul (son) - - - 14
In the supplemental section of the census:
Leybish s.o. Mortko - 24 y.o - lives in Staraya Ushitsa (Old Ushitsa)
Shlioma (brother) - 16 - died
Shlioma (1st cousin) - 16 - lives in the shtetl Brichany, Khotin uezd
I briefly looked through the pages of names in Staraya Ushitsa and found Leybish:
Leybish s.o. Mortko - 35
Aron (son) - 12
Mayorko (son) - 7
Shlioma (son) - 1