Thanks for all your responses. I believe coachBob is spot on when he says that the problem is relying too much on the kick to stay horizontal (and the fins are amplifying).
When I did the superman glide I noticed that the legs begin to droop slowly but I didn't pay attention to how much I needed to kick to prevent them from going down. I'll be sure to take note next time.
Also, the pool does have a deep end (10 ft) where I can try the vertical kicking. I'll put all other suggestions to practice and report back.