Front crawl: Hips sink when I breathe, causing bad things

My front crawl (according to my instructor) is GREAT. My feet stay at the surface. My hands barely splash. I do an "S" pull. I roll. But when I go to breathe, the following happens: 1. my hips start to sink 2. ... causing me to slow down 3. ... causing my head to sink 4. ... causing me to crane my neck 5. ... causing me to try to "grab" the water in front of me 6. ... causing me to nearly drown. Now we've tried a number of different drills and I've come to realize that my hips sink on almost ANY kind of rotation from front to back. If I do the TI skating drill, or even just a simple switch from front crawl to back crawl, or vice versa, my hips sink. Should I just keep plugging away at it, or is there some particular thing I should do? Kick harder when I rotate to breathe?