A date and place of birth for A.J. has been a brickwall for many a researcher. The few census entries that can be found for A.J. say he was born in 1833 in New York. However, Ancestry.com seems to have posted new millitary info and there's an entry of interest.
U. S. Army, register of Enlistments, 1798-1914
Booth, Ananias J., Hazel eyes, dark hair, fair complexion, b. 1829 Fishkill, (Duchess County) New York, Laborer, Registered 25 Nov 1850 (age 21) at Pokupsic (Poughkeepsie, NY) by Gibson, Infantry K, Deserted 12 Jul 1851, at camp in Fort ___________ (might be Fort Kaskaskia or Fort Washington).
Deserting the army would certainly be a good reason for A.J. to leave New York state.
I have A.J.'s military record from the Civil War. He deserted there too but returned in time for the Battle of Vicksburg. With the Civil War, getting away with deserting may not have been as easy as it would have beein in 1851.
The upshot is there's circumstantial evidence that would lead one to believe that the 1850 Army record is a good lead on A.J.'s birth.
I've written to Delta County, Texas (where A.J. died) to see if it's possible to find a death record. According to Delta County, death records weren't kept until the early 1900's. If I'm lucky, I'll find a probate or cemetery records. My next stop is the Delta County library to see what they can find. They ask for a small research fee. Not a problem. Any genealogist will tell you that genealogy ain't a cheap hobby.