It ain't the Weavers, but here's a story about how the music affected the politics.
Back in 1953 (note: this was before either the Army-McCarthy hearings or the Spureme Court decision in Brown v. Bd. of Education)a family friend had a barber shop in Portland, OR. At this time our life was mostly in the Negro community because me stepfather of the time was black, and so was his friend the barber. Anyhow, a guy from Lionel Hampton's band came in for a haircut, mentioned that the band would be playing New Years Eve 1954 at the Jantzen Beach Ballroom and that he'd get the local some tickets. Long silence ensued. The barber let him know that Negroes were not allowed at Jantzen Beach, at least not in the ballroom or swimming pool.
So they cooked up a plan. The band member arranged for tickets for 3 couple to be available (maybe 4) on New Years Eve. One of the band would hang out in the lobby, and when my folks and the other couples showed up he'd get word backstage that they were there. When the time came and the couples were in the lobby being refused admittance (even though they had tickets), Hampton stopped his band and announced to the management that they weren't playing any more until their guests were admitted.   There was quite a hooraw, my folks told me, but in the end the management gave in. And not only did Jantzen Beach get integrated, but (my mother insists) the band's playing improved greatly. Better audience feedback.

Direct Action gets the goods.

See, there were hundreds if not thousands of actions like that being taken all over the country in those days, by people who generally were never heard of. And those actions laid the groundwork for the big breakthroughs that everyone heard about. And people sang and danced and played or listened to jazz