Connect or "plug yourself in" to the lyrics of the song. Imagine you're sharing an important message with a friend. It's just a way of "talking," that's all! That takes some of the pressure off of the "performance" aspect. But if you are concerned about your performance/delivery/artistry, etc., just think of terms of "speaking" the message as clearly and as articulately as you can. Just as if you were talking to someone and you really wanted them to understand where you're coming from.
When I'm playing piano, I sometimes employ that mindset. I think in terms of crafting a "message." I'm NOT there to be judged; I'm simply there to be heard.
I'm writing this off the top of my head, but I hope it makes sense and helps!