Y DNA testing is perfect for a paternal parent search. This would enable you to match male line Rosenbergs precisely.
You should also consider autosomal testing for yourself and anyone else in your Rosenberg line. The objective would be to match with a broader range of other relatives: some will know where their families left. This should start to focus your search.
You can also hope that one of your matches is a good researcher with a reliable tree.
SHADUR, SADUR, SHADER, SADER, CHADOUR, SADOUR, SHADOUR, SZADUR, SHADDOW from Salakas, Lithuania
MINDEL, MINDELL from Utena and Vyzuonos, Lithuania
FELLER from Pabrade, Lithuania