His estate is liable. You may or not be liable. Even if you are liable, the debts may not be collectable.
It would depend on a number of factors such as whether you signed an admission form that contained guarantor language, etc. If you want to know your exact liability, you would need to take any and all forms you signed to a local attorney.
Whether you are personally liable also depends on what assets the decedant owned at death. The executors job is to pay all debts using the decedant's property. If there is enough property to satisfy the debts, then they must be paid and any left over property can be distributed either under the terms of the Will (if he died with a Will) or by the terms of the provisions of intestate succession (if he died without a Will).
Even assuming you are resonsible for those bills, whether they can be collected on is a different matter. Texas is a debtor-friendly state with respect to unsecured liabilities. Medical debt is an unsecured liability. That means that even if you are responsible and the creditors get a judgment against you, they can only "foreclose" on non-exempt assets. Exempt assets include your homestead, up to $30,000 in personal property, wages, certain retirement assets, etc.
So when you go meet with an attorney about this, be sure and ask both questions: Am I liable, and if so, are the debts collectable.
FINALLY: Some debt collection agencies will try to pray on guilt and grief. Don't start paying debts until you've had chance to review all of this with an attorney. Some collectors manage to get the bereaved to transfer the debt into their own name by use of a credit card. Don't fall for this. Invest in a consultation with a probate attorney and take further action AFTER that meeting.
I am sorry for your loss and I hope the medical debt issue resolves without compounding your pain.