Hi. I had trouble replicating your immigration findings. I wanted to see the actual manifest. I found it under Josef Zawodsky. The image is attached. Not much information there. I do not see Josef's father Franz. Instead, I see two children: Josef 7 months (on your list), and Franz 3 years. If they had another Frank born in 1883, then this one must have died.
I searched for them in the censuses AND FOUND A POSSIBLE MATCH IN MADISON COUNTY, NEBRASKA! At this point, one switches to the Marge Sobotka book, Nebraska, Kansas Czech Settlers 1891-1895 and looks for the data.
On page 112, Jos. Zavodsky is from Samrakov, Telc, Jihlava, Moravia. It was a common (albeit annoying) practice of census takers to put down Bohemia even when the person was from Moravia. With them coming from the Jihlava area, the records may be available on actapublica.eu. Give us a few days; we might find a match for you. Samrakov is probably misspelled, but people who know the area or have a good map will be able to look near Telč and figure out which one it was. But it also looks like Telč is very near Bohemia...
If this is NOT your family, reply here so we don't go looking at the wrong books. :) I was not able to find them in the Nebraska 1885 census. There are two Zavodskys in Kansas: Frantisek from Brandyn, Dacice, Jihlava and Anna from Lomnice, Trebon, Budejovice. Reference is made to another Frantisek (Frank) who lives in Kendrick, Idaho.
The 1900 census says the Josef in Madison County was born in March of 1848.