This article goes in-depth on members-only unions, especially in the South. Apparently, they’re far in between, but they exist and are often very civil rights-oriented. Read this:
What is often lost in many of the discussions on workers’ rights is that members-only unions are not a theoretical construct or historical remnant. In fact, beyond UAW Local 42, a variety of public and private-sector locals have operated on a members-only basis for many years, with varying degrees of success. For example, the American Federation of Teachers (AFT) has approximately 120,000 members in members-only unions spread across Texas, Utah, Arizona, Colorado, Louisiana, Mississippi, Alabama, Georgia, North Carolina, Tennessee, West Virginia, Wisconsin, and Virginia.
Most of the existing members-only unions are located in southern states, because legal conditions in those states such as right-to-work laws make it difficult to organize a majority union. Similarly, most members-only unions are public-sector unions, because many states that are inhospitable to labor can easily pass laws that limit collective bargaining rights in the public sector. However, there are members-only unions in the private sector and in other geographic locations as well.