Hi Jeff -- thanks for posting to History Hub!
Historical examples of district newsletters can be found in the personal paper collections of Members of Congress.
Personal paper collections are not held by the National Archives. Unlike the official records of congressional committees, personal papers (like newsletters) remain the property of the Member. Many choose to donate their personal papers to an archival repository.
The best way to locate personal paper collections is through the Congressional BioGuide.
The Center for Legislative Archives also maintains a congressional collection index. The index is arranged by state, and then by the name of the holding institution, followed by the name of the Member of Congress.
For more background on franking, check out all the Congressional Research Service reports on the topic.