The term "public record" usually refers to records that a government is required to maintain and make accessible to the public. In most states, I believe SMS messages sent by government employees while conducting official business are considered in-scope (though there may be other laws that make certain messages private).

As far as whether your SMS messages can be accessed by law enforcement without a warrant, it gets more complicated.

Older than 180 days? Fair game. A court or government agency can subpoena your provider without any requirement to notify you, per the Electronic Communications Privacy Act of 1986.

Newer than 180 days? A warrant is required, but there are exceptions. For example, administrative subpoenas are allowed in cases of national security.

Records of incoming/outgoing calls require only a subpoena, same with cell tower geolocation data and IP addresses. However, wiretapping of actual calls requires a warrant.

The reality of how and when the government accesses your data in the real world is probably different than the laws as written, so of course take everything with a grain of salt. The best solution with text messages is at least E2E encryption so that the provider cannot store them. (Accessing messages stored on your device itself does require a warrant).