I agree that girls were often sent off "somewhere" to have their babies in secret and then return home as if nothing happened. I remember this happening to a girl I went to high school with and what better place to do it than Houston?
As far as Gulf Coast Placing Bureau goes, I see they are a defunct corporation, once having changed their name to Gulf Coast Children's Aid and Protective Society (see:
http://www.corporationwiki.com/p/hl1v5/gulf-coast-childrens-...). Someone or some entity must hold these records...they couldn't have just thrown them in the trash. I looked on the internet and couldn't find a repository for the records but I know that like funeral homes, places such as these don't typically just roll of the sidewalk but sell out to someone else. But to whom is not obvious at all. You might want to try calling the main branch of the Houston library and ask for their History or Genealogy section and hope to get someone nice and helpful. You could also try the Clayton library in Houston. Tell them/ask them if they don't know then to try to give you a lead to somewhere else you can call. You might even want to try calling the Secretary of State that set up the original corporation. I did a search on a subscription newspaper site and found reference (only 1) to Gulf Coast Children's Aid and Protective Society in a 1931 article that was insignificant but named Sarah Mathews as the director. That means the name was changed a long, long time ago and that's the name you really need to look for.
As far as unsealing goes I'm not sure how this would work for you because I assume you are not the person who was adopted but the child of that person (or other relative) and that may be handled differently. There are procedures for this and some guidelines may be found by googling "unsealing texas adoption records" and then looking for "Texas Adoption Information & the Central Adoption Registry".
Who is Mattie Lee Hood...I thought we were looking for Frankie Lee Hood?