I don't know anything about Texas non-compete law, but "right to work state" refers to restrictions on unionizing. It has nothing to do with non-compete clauses.

The "right to work" is about restrictions on making unions mandatory, so you are correct but glossing over an important details. In practice it generally means that forming an effective union is much harder and thus harms workers, but theoretically it could be paired with other laws that supported unions but that didn't centralizing their power and removing worker choice.