How can I avoid becoming wrapped up in family issues and control my anxiety about them?

Based on the information provided, this is an issue with boundaries.   Sounds like your uncle goes to your mom and possibly your mother takes the issues to you.  I do not like to assume so I will direct you to information regarding families dealing with enmeshment and triangulation.  Enmeshment is the concept of diffused boundaries in families and other personal relationships.  Triangulation is two individuals relaying information to a third party what cannot be said directly to each other.  From someone who has dealt with both these issues on a personal level, being informed about family dynamics is beneficial and can help you develop healthy boundaries for yourself within your family structure.  You may need assistance with developing healthy boundaries.  Although it is possible, it is not easy to change the way you have been operating for any extended amount of time.  Hope this help!