Born in Texas, Adopted in Oklahoma

Here is my dilemma. I have moms name only, along with babys name and birth date when born in the Virginia K Johnson Home in Dallas 1927. That's the only information I have. The baby was handed over to adopted parents in Dec 1927 and then adopted in Oklahoma January 1928. If I contact Austin Records, will I be able to get any information on birth parents from the records that would have been kept in the Home? even though the actual adoption took place in another State? The mother signed over parental rights to the Home 6 months before the adoption took place. Time is passing and although family would be long gone, the medical and other family information is needed.