Have you searched for the 2 boys under McGraw/McGrath?
Yes, and also searched the Roman Catholic Records on Findmypast.
What about tracing that family for clues to Rose and her children?
I presume you mean the "McGrath/Mcgraw" family. I have searched backward and tried to find a connection between that family and Rose Ann's family, the Flanagans (e.g. was Rose Ann's adopted mother Mary once a Flanagan?) The closest I can come is a Rosanna Flanagan born 12 Oct 1876 to James and Mary Ann (nee McCormick) Flanagan (
https://goo.gl/LW8aFV) and the deaths of (her parents?) James in 1877 and Mary Ann in 1876. So also looked for a connection between McGrath/McGraw and McCormick (including variations), to no avail.
But that is a different path/story, and the focus here are the births of John (c.1903) and Robert (c.1906).