The student's sentence does not mean what they intend it to mean. It means that "her" songs, as if they were a person, are physically touching the soul, the same way that I feel my keyboard at my fingertips this very moment.
Sentences A and B make sense and are natural. For sentence C, people should be able to understand the meaning, but it is very unnatural.
Sentence A is by far the best and most natural. In this case, "touch" means that it emotionally moved the student. People use "touch" like this very often. "Her songs were very touching," meaning her songs really moved me emotionally.