Erica, thanks for the Eagle Pose... just off to try it (I love exercises like that!)
So far as singing with eyes closed is concerned, I do have a couple of cautionary tales to relate:
1. A friend was once singing with eyes closed, and shuffling feet at the same time... When he finished singing and opened his eyes he was surprised to find himself facing the back of the stage. Embarrassing for him, but the audience loved it!
2. Once when I sang Love is Pleasing, (swaying as I sang) I opened my eyes at the end of the song to find the entire audience (around 60) standing in a circle around the room, arms linked and swaying, and insisting on a repeat performance. I believe we sang it through twice more, for good measure.
So now I take a peek if I think there's a chance of any hanky panky. And I keep my feet firmly planted.