No — it is not filed with the Clerk, and there is nothing to confirm with the county. Texas has no state-issued Declaration of Domicile and no county recording process, so neither you nor SavvyNomad submits it anywhere. There is no filing to look up, and no need to contact the Aransas County Clerk's Office about it.
The declaration we provide for Texas is a private statement of your facts and present intent. Keep it with your personal records.
Notarizing it does not mean that Texas DPS, a tax agency, or an election office has approved your residence or domicile. It is simply for your own records.
Completing it and finding it later
This step is optional. You can complete your declaration here: https://app.savvynomad.io/home?tab=To-Do&step=DoD
Once it is done, your completed declaration is available in the Documents tab of your portal, where you can download it and store it with your own records.
Separately, your declaration is independent from getting your state ID or driver's license — you will not need it at your DPS appointment.
For more on why the two states differ, see Why do Florida and Texas use different Declaration of Domicile processes?