I took a quick look at the Texas law. Like a few other such laws it allows sites to use an external service to do the check, as long as the service uses a "commercially reasonable" method of doing that. That basically means it has to be based on government ID or by inference based on certain types of transaction records they can get access to (e.g., if you have a mortgage they can reasonably infer you are an adult).
As far as I can tell it would be possible to build an age verification service based on an open source ZKP implementation such as Google's Longfellow [1] that would be acceptable to these laws, but would allow anonymous age verification. It would be similar to the system the EU is now trialing, except not limited to iOS and to Android devices with Google Play. Longfellow should be able to work with those but also most modern smart phones running any OS the supports the phone's secure element, and also desktop computers that have secure elements, and devices like YubiKeys.
You would have to verify your age with the age verification service to set things. The easiest way to make it so that is not a privacy risk is for the age verification service to be offered by some entity that already has your ID documents. In the EU that would be the governments themselves, but I don't think any US state governments are ready to do that.
The age verification service doesn't necessarily need to store copies of whatever ID you present. It just needs to know you are when it issues its ID documents that get bound to your device's secure element. If this service was offered by some entity that has a widespread physical presence (a bank would be perfect) you could go in, show ID in person, and get your device enrolled.
Even better would be for a trusted non-profit to run this, like the EFF or the ACLU. Yes, I know they don't want age verification to happen at all, but they are going to lose that one, and it would be prudent to try to make it so that people have a privacy preserving way to do it that can be used anonymously when that happens.
Anyway, once your device is set up verifying your age to a website would involve a protocol between your device in the website the uses a ZKP (Zero Knowledge Proof) to demonstrate to the website that the identify information the age verification service bound to the secure element on the your devices says your age is acceptable. The ZKP doesn't disclose anything else from your identidy information. (The web server sees your IP address of course, but they would see that without age verification too). Note that the age verification service has no idea when, or were, you age verify at a website.
[1] https://github.com/google/longfellow-zk