How can I find information on someone when I don't know what area of Texas they moved to?

Have looked all the usual places online for a very long time, and need some advice, probably from someone who knows more about Texas than I do.

I am curious to find what happened to William Gazzam the 3rd, born in England 1788, moved to USA with family as a teenager, served in war of 1812. Family book says he moved to Texas after 1814 and died there leaving behind a wife and children, but I have not been able to find any records for them in TX, nor any of their names. 1850 census no help so far. If you have advice on where to look next, please let me know!