Joe,
William E. Barry also listed as William A. Barry is listed in the 4th Texas, there is no one named Berry in this regiment. There are no other persons named Barry, from Texas, listed in Fold3.
In the 4th Texas State Troops there is an Anderson Berry and an E. S. Berry but no William.
The following are also listed on Fold3:-
… Tenth Cavalry (Locke's Regiment) › B › Berry, William
… Eighth (Hobby's) Infantry › B › Berry, William T (29)
… Tenth (Nelson's) Infantry › B › Berry, William E D
… Fifth Infantry › B › Berry, William H (25)
… Nineteenth Cavalry (Burford's Regiment) › B › Berry, William C (31)
… Tenth (Nelson's) Infantry › B › Berry, William W (30)
… Seventeenth (Consolidated) Dismounted Cavalry › B › Berry, William W
Hope this helps.
Alan Thrower
England