An 1879 History of Lorain County Ohio, has a couple of brief references to an Adna Warner in the history of Columbia Township. The information is only fragments, as compared to a consise family history, but this is what I found. By the way, is it possible that Adna's wife's maiden name was Bronson, as opposed to the Brunson, listed in your posting? I mention this because as you will see below, the name Bronson was very prominent among the early settlers of Columbia.
Under the heading "Purchase", the history indicates that Levi Bronson,Azor Bronson, Harmon Bronson, Calvin Hoardley, Jared Pritchard, and some 15 others formed an association called the Waterbury Land Company, which in 1807 purchased the land later settled. The association was named the Waterbury Land Company because the settlers were all from Waterbury Conn..
Under the heading 'Settlement", the history indicates that in Sept. 1807 a company of 33 persons left Waterbury Conn. for Columbia Township. Among these people were Bela Bronson and family, and a Chauncey Warner.
A little further in the history, it says "in 1809 Noah Warner joined the settlement".
Later still, the history says "Adna Warner bought out Benoni Smith, on lot 50, and became a settler in 1814". (Benoni Smith was one of the original settlers in 1807).
Under the heading "The Militia", the history indicates that in 1810 a militia company, composed of men from the townships of Columbia, Ridgeville, Eaton, and Middleburg in Cuyahoga County, was organized. Calvin Hoardley was commissioned as Captain on Oct.25, 1810. Among the members of this militia company were several men with the surnames Bronson and Warner. They were:
Daniel Bronson Bela Bronson Levi Bronson Noah Warner Chauncy Warner Adna Warner Aaron Warner
The fact that Adna was apparently in a militia company in 1810, would indicate that he was in the area prior to purchasing the land in Columbia in 1814. Is it possible that he was the son of one of the other Warners mentioned in the history?
As a final note there are three young children buried in the Columbia cemetery, who are listed as children of Elmer and Elizabeth Warner. The children all died in the 1840s and 1850s. They may be children of Adna's son Elmore and his wife. The information on the cemetery inscriptions can be found at the Lorain County GenWeb site. Hope some of this information is helpful.