And for what it's worth, some city directories. I started at 1878 trying to match the 42 Cherry St address you say they were at on the 1880 census, but didn't find it. Also from 1892 and worked backward, but didn't quite meet in the middle. I didn't do whatever is available from 1884 to 1888.
New York city directories
1878 City Directory
William Sullivan, h 110 South Street probably the same man as 1889
no other Sullivans at that address. no Sheas or O'Sheas at that address
1879 City Directory
William Sullivan, h 3 Canal St
no other Sullivans at that address. no Sheas or O'Sheas at that address
1880 City Directory
William Sullivan, clerk 519 W 28th
1882 City Directory
William Sullivan, clerk, 408 E 85th
1883 City Directory
William Sullivan, clerk, 408 E 85th
1889 City Directory
William Sullivan clerk, h 110 South probably the same man as 1878
William Sullivan clerk, h 18 Varick
1891 City Directory
3 Wm Sullivan, clerk
115 Washington Place <-- likely him
890 Boulevard
1384 3rd Ave
1892 City Directory
William Sullivan, clerk 18 Beach <-- him
other William Sullivans at
890 Boulevard
1384 3rd Ave
122 E 113th
311 W 13th
?? Jeremiah O'Shea at 26 Beach
?? Margaret O'Shea, widow, 22 Beach
Assuming that is him at 18 Beach in 1892, then that eliminates 2 of the 3 addresses in 1891, because those men are still at the same addresses, leaving 115 Washington Place as the only possible address for your William in 1891. - assuming he was listed in the first place.
Best bets are the later ones at 18 Varick and 115 Washington Place. And the 18 Beach Street address is where the child Elizabeth Sullivan was born in 1891, daughter of Wm Sullivan and Catherine O'Shea.
I have my doubts about the earlier addresses, but included them anyway.