Is there a link to a tune for this anywhere?

Songs like this are always potentially religious, and always potentially political. Even when the people singing them don't think they are. They speak to the place where the two meet, which is the human heart.

Would it make sense to say that "We Shall Overcome" shouldn't be seen as religious or that it shouldn't be seen as political?

Sticking overtly religiose words into somgs like this isn't necessary and can get in the way of the real message. But that's a diffeent matter.