I've spent a lot of time searching for information and clues (as have several others here) and it's a bit disappointing that Dorothy doesn't seem interested in finding living relatives (see reply #108). I know that another Rootschatter had previous found other relatives for her but not sure if any contact was attempted. It's possible, of course, that by now there's no one left alive who known the whole, or even part of, the origins of Eveline but DNA testing might be useful in confirming or ruling out any links.
It seems as though David Shulman had only 2 sisters who could have been Eveline's mother. One thing that hasn't been questioned so far is who actually raised Eveline? If she's the adopted child in 1911 census (as she probably was) then did she stay with Annie after David died? Religion hasn't really been mentioned on this thread but as the Shulmans were Polish Jews and Annie O'Brien was likely Irish Catholic then the differences in religion and culture could have caused problems.
Here's a very brief outline of Shulman family so far (more details to follow after groceries unpacked, hovering done, etc.)-
Benjamin Shulman (c1862 Poland/Russia-1931 Manchester No.) m. Dina(h) Barrett/Baritz (c1857-1897 Canongate); m.(1898 Newcastle) Minnie Solomon (c1872 Poland/Russia-1916 Prestwich, Manchester)
1. David Shulman (1883-1921 Eng.) m.(14 Dec.1908 St. Machar, Scot.) Annie O’Brien. She m.(16 Apr.1921 Chorlton) Bertram Johnson (1892-1964)- he m.(24 Dec.1938 at Ashton under Lyme) Phyllis Miles.
2. Samuel/Harris Shulman (1886-1923 Liverpool)
3. Hannah Shulman (1888 Cheetham-1891 St. George)
4. Dora Shulman (1890 St. George/Edinburgh-aft.1911) registered as Rebecca Shuleman
5. Rachel Shulman (1892 St. George/Edinburgh-aft.1911) registered as Shuleman
6. Solomon Barnett/Bernard Shulman (15 Sept.1894 St. George/Edinburgh-11 Nov.1950 New Rochelle, Westchester, N.Y.) m.(1922 Liverpool- as Barnett S. Shuleman) Sarah E. Sondak/Simon
7. Eva Shulman (1897 Canongate-1898 Newington)
8. Esther Shulman (c1899 Edinburgh-aft.1911)
9. Sarah Shulman (1900 Edinburgh-aft.1911)
10. Abraham Arthur Shulman (1902 Cheetham) m.(1934 Salford, Manchester) Rosa Michalovitz Raffles