"Republic of Texas: Poll Lists for 1846" compiled by Marion Day Mullins lists:
White, Carey - Washington Co. White, Anderson - Washington Co. White, Harden -- Washington Co.
I know White is a common surname, but all of these men have less then common given names and I thought they might be related.
If you go to the Handbook of TX Online and type in Carey White, one item comes up, a brief bio onAlbert G. Gholson and it states he fought successfully under Capt Carey White.
I would look for a published Washington County history, early settlers are usually at least mentioned and any man with military connections is usually mention, too.
Have you checked the Washington Co, TX GenWeb site?Some GenWeb counties are putting their local records online.I would post your query on that site, too.
I did a very quick check and I see the 1850 census is on there with C.M. White 48 b TN; Anderson White 31 b TN; Harden White 34 b TN.C.M. is listed with N.M. and assume that is Nancy and all the children are initials, too.
On the list of volunteers who will do look ups is someone who will do cemetery lookups, you might email them.