Name Change on COVID Vaccine Card in Texas

Hi all! I decided to take my husband’s name when I got married (because Smith is the worst lol) but one of the last things I have left to change is my COVID vaccine card. We are planning a lot of traveling next year so I don’t want there to be any issues with it not matching my new legal name. I can’t seem to find anything about what the rules are in Texas for who can rewrite your card. I won’t need to go back to where I got my primary doses, will I? I was very desperate to get vaccinated as soon as possible because my sister is immunocompromised, but in early March 2021 the only town giving out vaccines without an age or disease requirement was 2.5 hours away from Houston, so that’s where I got both primary doses. I really don’t want to have to drive all the way back there just for my card to be rewritten! Has any other recently married Texan on here gotten their card rewritten and can advise?