Can I take Affidavit of heirship, and death certificate to the bank in order to access my
deceased mothers bank records so that I can see what she was doing prior to her death in Texas?
In practice, it's up to the bank. The banks usually won't accept them, as it creates liability for the bank. If you really want to find out what happened, then you would have to apply for letters testamentary (if she had a will) or letters of administration (if she did not have a will.) These procedures will allow you to step into your mother's shoes and do anything she could do, such as ordering bank account records, etc.