A quick web search turned up several groups with that name.
I didn’t see one that matched your description.
There was a “Crossroads Quartet” back in the late 1940s and into the 1970s that was part of the Arthur Smith radio and television programs. His country music show was syndicated and appeared in many areas of the country. The quartet closed each of his programs by singing a hymn or gospel song. They participated from time to time with the Billy Graham crusades. The members of the group were Arthur Smith, “Brother Ralph” Smith, Carol Honeycutt, and Tommy Faile. When Carol left the group she was replaced by Lois Atkins. I have some of their recordings, but not the songs that you mentioned.
Arthur is a multi-talented individual. He plays numerous instruments, and plays them very well. He was deeply involved in television production for many years and owned a recording studio. His ability as a song writer is well known. He wrote “Guitar Boogie” and “Feuding(Dueling) Banjos.” He wrote a number of gospel songs that were recorded by Johnny Cash, The Harvesters, The Prophets, The Statesmen and other gospel quartets. Some of his songs include: “I Saw A Man”, “The Fourth Man”, “Not My Will”, “I’ve Been With Jesus”, and “He’s Everywhere.”
The Crossroads Quartet often performed these songs that were written by Arthur.