Ron, You've taken quite a load because of the limited information you have.
The Texas Death Index and the SSDI show William Henry WALTERS b. 10 May 1912 and d. 11 Sep 1996 in Grimes County, Texas
There are also twoWilliam H. WALTERS who both died in Dallas County but their dates of birth were much earlier than 1912.
There were numerous "William WALTERS" that could possibly be the one you are searching.And of course he may be still living or he may have died outside of Texas.
I think the one that died in Grimes County has enough merit to try and find out more about him.I suggest you try to get a copy of his obituary which will should be in the Navasota newspaper and is probably archived in the library at Navasota.If you don't live close enough to Grimes County to visit, go online to the TxGenWeb Grimes County web site and see if there is someone listed who does obituary lookups for Grimes County.If no one is listed, put a message on their message board (Query forum) asking for someone to look for the obituary and send you a copy.There should be enough information in the obituary to determine if it is the right 'WALTERS'.
You might also go online to one of the "white pages directories" (I use www.switchboard.com frequently) and get a list of all the funeral homes in the county.A letter or telephone call to each one will probably result in locating the one that handled the burial.Most funeral homes keep a copy of the obituary and are very cooperative about providing a copy.Don't forget to include a self addressed stamped envelope.