Davya, the earlier suggestions to look for death certificates and passenger manifests are really excellent.

Regarding other spellings of the name, I had a few thoughts.
1) I'm assuming that your family came from Eastern Europe rather than Germany or Austria. I haven't seen any Slavic language that uses a double "o" in the middle of a word. I think the original spelling is more likely to have a "u" instead of the double "o". Another possibility is an "o" with an accent (ó), which would also have been pronounced like the English "ooh."
2) If they lived in Poland or in areas in Belarus or Ukraine that might have been part of Poland at the time, the ending "tsky" was almost certainly "cky" as the letter "c" in this word would have been pronounced like "ts" in English.
3) Also, and this is really a long shot, if at some point the name used the Polish spelling, the second letter could be different. The original may have started with "Sł" where the "ł" is pronounced like a "w" in English. More important, when indexing is done by people who don't speak the language of origin, this "ł" is often transcribed as the English "t," and so you might also keep an eye out for names starting in "Stu" in the indexes of passenger manifests or other documents indexed by Americans.

Mary Ann Evan