Sorry for the late reply.
Your definition of Logorrhea does not meet mine.
Mine is expressed by excessive wordiness with minor or sometimes incoherent talkativeness. Logorrhea is sometimes classified as a mental illness, resulting in a variety of psychiatric and neurological disorders including aphasia, localized cortical lesions in the thalamus, mania, or most typically in catatonic schizophrenia.
Humming lyrics or tunes does not fit with the above.
As I see it, a psychologist, not a psychiatrist, would be my referral.