I have just spent over an hour looking at census records on Ancestry.com. In 1850, the first year every person was listed by name in the US census, there are 1,544 women and girls listed as Sabrina, and 1,644 listed as Sabina, in the index to the censuses. Of course the problem is that both census takers and the people who indexed them made mistakes, so I tried to find a woman who was definitely listed as Sabrina in two successive censuses. That turned out to be hard to find, but I think I managed it with a woman called Sabrina Thompson, living in Summit County, Ohio, in 1850 and 1860. She was the wife of Hiram Thompson, and was born in New Hampshire around 1806. Interestingly, the index to the 1850 census lists her as Sabina, but when I examined the actual census record it looked more like Sabrina to me.
So Sabrina existed in the 19th century, but it was rare -- so rare that census takers seem to have frequently confused it with Sabina or misunderstood it completely. Another example I found was Sabrina in one census but "Subline" in the next. Subline is so odd that I think it's likely the woman's real name was Sabrina, but we can't be sure without further investigation. :)