Barbara! WE HAVE A MATCH.
I have a William J. Brown in my tree as follows:
My GGG GF was John Brown, b. 1794, NC. Wife Elizabeth ?, b. 1823, SC.
I believe they moved to Dale Co., AL around 1840.
Children were:
Jesse D. Brown, b. 1840
Rebecca b. 1842
*William J. b. 1844 (I think this could be your William as well and I have no other info on him so your info of moving to TX may be why I could not find him after 1860 in Dale Co.)
John G. b. 1848
James R. b. 1850
Daniel A. b. 1852
Mary E., b. 1857
If you think this is a match I would be glad to share all my brown info and would like to update William with your info.

Norm Brown