Kearns/Karnes/Kerns families surnames were often mis-spelled and are hard to sort out. By about 1800-1825 there were at least 3 different families living in Monroe Co. (formerly Greenbrier Co.) and southern Greenbrier Co. They apparently were a German family near Alderson, an Irish family near Lewisburg, and another German family near Peterstown in Monroe Co. The "Elizabeth Kearns" you mentioned was possibly a sister or cousin of "Alexander Kearns", and they seem to have descended from the Irish family.
There was a Will of Abraham Kearnes in Greenbrier Co. 11 Jul 1828, which named his wife Elenor and 2 sons named Abraham & Thomas. Old Wills seldom named all the children, but often named a wife and 2 older sons to ensure the inheritance of property. Elizabeth and Alexander were not named, but might have been younger children. While I am not researching the Irish family, and don't know what happened to Elizabeth, I ran across some information about Alexander that might be helpful.
Alexander Kearns (c.1794 - 11/18/1881) married Elizabeth Hanna 12/23/1817 in Greenbrier Co. They moved down into Monroe Co. and bought several tracts of land northwest of Union. They had one daughter named Elizabeth (b.1838). Alexander's wife (Elizabeth Hanna?) may have had childbirth complications, and she died 10/10/1838 and was buried in the Green Hill Cem. in Union, WV. Alexander sold his properties in the 1840s and moved back to Greenbrier Co. He re-married a woman named Nancy. The 1850 Census (in Greenbrier Co.) shows Alexander-56, Nancy-42, Elizabeth-11, and an older woman named Elenor Karnes-86 who was born in Ireland. (This may be the same woman "Elenor" who was named in the 1828 Will of Abraham Kearns, and quite possibly she was Alexander's mother) On 30 Dec 1857, Lizzie S. Kearns (Alexander's only daughter) was married to Franklin Montgomery. Alexander died 11/18/1881 and was buried in the Old Stone Church Cem. in Lewisburg, WV. His death record said he was 87 years old, and had been born in the "Irish Corner" community of Lewisburg, WV. Alexander's Will mentioned a niece named Elizabeth Hanna. Since your Elizabeth Kearns-Hanna died in 1850, this might have been one of her daughters.
A friend discovered all that information on Alexander Kearns while we were researching the German Karnes/Kerns families. We weren't even looking for the Irish family, so there is probably much more information available within the old records of Greenbrier Co. Good luck with your research!