I'm not an expert on the rings, but have experience with the archaeology. The fastest way to answer your questions is to contact the Office of the State Archaeologist. Texas is massive, and it appears to be divided into seven geographic regions, so you will want to contact the correct office for your region with questions.

That said, unless Texas law is fundamentally different from the other states I've lived and worked (very unlikely), this is private land and an archaeologist won't force their way onto your property screaming, "This belongs in a museum!" The State Archaeologists are more concerned with preserving sites in immediate risk of destruction, like those in the way of a new road or building construction, and maintaining known sites on public land. The official handbook for the Texas State Archaeologist says

The office conducts field surveys to record and evaluate sites, with priority given to those in imminent danger of destruction. Services provided to the public include site examination, conservation advice, and lectures.

So if you contact them they may be able, with the help of a few photos and your GPS coordinates, to give you a brief overview of what you're looking at, as well as your legal rights regarding found items.

As an anthropologist I of course urge you to be a good steward of the materials on your land. This means calling on experienced archaeologists to excavate instead of digging willy nilly yourself, and providing associated descendants with the choice to reclaim their cultural material, or a local museum to curate the material so others may enjoy and learn. Depending on your location the rings may be associated with the Comanche, Caddo, Apache, Kiowa, etc. These are vibrant nations who value their history and connection to the Texas landscape. They, like you, are an integral part of Texas history. It may fall to you, here and now, to decide what becomes of this aspect of their material legacy. I sincerely hope you choose to preserve that history to the best of your ability.

You may want to ask this question of /r/Askanthropology since there are a lot of archaeologists over there. There are some here, too, so you may receive some information with this post.

For the most part, federal law protecting historic/prehistoric sites only kicks in when there is federal involvement. Since these are on private land, they generally would not be affected by federal law, but if you were to reach an agreement to put a wind farm on those hills, for example, there might be federal permits involved that could require survey and mitigation.

If there were human remains associated with these sites, federal law might be kicked in, but even in that case, it is not likely to be an issue.

All of this leaves your questions to be answered by someone who knows Texas law (and I'm not that person!). My guess is that a state like Texas will give a great deal of latitude to the private property owner, and I doubt very much that there is any provision for eminent domain in the name of archaeology. In general, archaeologists are stretched too thin and are trying to address a few sites that are easily addressed - bureaucratically and economically. They aren't in the business of picking fights with local property owners (except when federal permits are involved - and that situation is driven by the federal agency), so I believe you're safe in that score. But there maybe someone more familiar with the situation in Texas on this or the other subreddit who can answer your questions on these scores.

Somesimes local university archaeologists will be interested in a site like this, so I wouldn't hesitate to contact them to see. There is also the Texas State Historic Preservation Office, which can answer a lot of your questions and may be able to direct you to someone who might be interested. There are likely to be artifacts associated with this site. Artifacts are like pages in a book, best understood in the context of the book. Artifacts taken from a site are like pages ripped from your local library book. The pages tend to have no value once they are torn from the book, and the book on the shelf has less value. So I encourage you to leave things as they are, if you want the site to contribute to an understanding of your state's heritage and prehistory.

Great questions on your part; best of luck with dealing with this resource. It sounds interesting.