I finally did it. For almost 20 years I have been thinking about how to reconcile Psalms 81 which specifically says to blow the Shofar on new moon with the traditional Karaite view that there is no commandment to blow the shofar on the Yom Teruah (i.e., what the Rabbanites call Rosh Ha-shanna), which is a new moon. And this weekend, as I was preparing for a class I taught at a Conservative Synagogue, I did it.
It was actually pretty easy once I read the entirety of Psalms 81 (and not just the part that was perplexing me).