I think I remember reading in Rabbi Simcha Bunim Cohen’s sefer on the subject that one should not sing during the Nine Days (except on Shabbos).
As far as listening to your own recorded singing, why would that be any different than listening to other people’s? And yes, rabbanim have written that one should not listen even to unaccompanied singing.
Hey, if your songs have original lyrics (that is, you wrote them), would you be willing to post them?