i have copies of some pages of francis berkshire's excellent book, and it gives quite a lot of information on ralph lazier berkshire. ralph did settle in henry county indiana, in newcastle. he had a daughter named elizabeth, but she doesn't quite fit yours. this elizabeth r. berkshire married james b. wright on 8/25/1830 and your elizabeth, born 1818, would have been too young to marry.
francis does note on page 57 that there was an elizabeth barkshire who married john allee 5/13/1841 in henry co., and a sarah j. barkshire who married elihu ridgeway on 12/24/1848, but francis notes that it is not known whether these ladies are related. further, he notes, there was an 'anson berkshire' born 1798 living in henry county, wayne twp in 1850 who was born in ohio. his wife jane hart age 38 was born in NJ. a son william, 17, was born in ohio.

the name spelling (berkshire, barkshire) is not important, since the census takers were often pretty free with spelling and those who were giving the information often didn't read/write either. there's a possibility that your elizabeth is related to ralph berkshire's tree somewhere but it's not apparent. if you can find an 'anson' berkshire in trumbull county ohio, then that would be a lead for you.
good luck!
tanya