I use bilateral breathing. When I breathe on the left side I can feel my body angling in the water so that my feet sink. Of course this makes me slow down. What is causing this? Am I lifting my head? I don't *think* I'm lifting my head too much because if I lift it any less I end up sucking in some water. Advice?

I have the same problem. Would a possible solution be to kick harder/stronger?

You may be taking breaths on your left side slower than on the right side. Pay close attention to your stroke rate during different breathing patterns and see if you're strokes on the left are slower.

If it's not a speed issue then perhaps you're a little less flexible in one of your shoulders.

it might be that you are not rolling your body enough, remember that swimmers don't swim on their stomach. That was my problem when I first started out, also I just favored one side more then the other because it was my stronger side. Keep working at proper technique and you'll get stronger and it won't even be a problem.

The other thing could be your pulling motion under the water. When swimming normally, your hand should form an "S" through the water. Attempting to "reach forward" after hand entry helps with shoulder/torso/body rotation.

During breathing, what I have seen some people do is lose track of the center line, often crossing over with the hand/arm opposite the side they are breathing on. Meaning the torso overcompensates for the breathing effort. This causes the pull stroke to lose forward momentum and cause the swimmer to sink a bit. I'm probably having a real hard time explaining... If you are breathing to your left, make sure your right hand isn't pulling toward your left when starting the pull.

One technique to try is to keep the water line when breathing between your goggles. The head ends up making a small pocket to allow breathing without inhaling of water.

Any of this sound like something that might be occurring?

It is important to keep your head down and raise you hips up to the surface. This allows your body to flow through the water more easily. In a longer swim you really should keep your head down like you are looking at the bottom of the pool.
If you feel like your body is sinking it is probably because you are taking a shorter stroke on the other side, you are lifting you head, or you are letting your hips sink. When I was taught how to swim I was told that it is important to bi-lateral breath as well. I think it is important to keep your body balanced in the water but I don't think it is neccesary to breath to both sides. I've been swimming for my entire life and all through college and I found that if you are stronger breathing to one side then you should just stick with doing that. If you watch all the Olympians, specifically Phelps, Lezak, and Vanderkay, you will notice that they only breath to one side during their entire race.
Just something to think about.