You should speak with a Texas attorney. The laws of each state are different. Since your mother passed in Texas, Texas law governs. That state may have a similar process as we have here for small estates, which is to provide an affidavit of small estate to confirm you are the successor and no probate is or will be pending for your mother's estate. Good Luck.
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Most states have an abbreviated procedure that allows you to claim the funds through a relatively simple filing or an affidavit if you are the only heir and the estate is small. A TX attorney will have to help you with it though.
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I agree with colleagues. Most states do have abbreviated process for administering a small estate. Most banks will require court order to close account based on facts you provided. In your interest to act before funds turned over (escheat) to state. Consult Texas probate attorney.