Total Immersion

I used TI to develop my stroke.  I'm no expert, but if you go to the official TI site and post a question, there are tons of TI certified coaches and experts who are super helpful and will respond quickly. 

http://www.totalimmersion.net/forums

I am a novice at TI.  I started with it 3 mo. ago.  I read the book.  The drill that has you swim on your side is really helpful for what your asking about.  It is important to do this with fins on as it allows you to keep your speed on your side while pressing your buoy.  Also, It took me 2-3 sessions prior to feeling comfortable doing this.  Now I'm like a fish, I swim lengths on my side with no problem.  Having done these drills, now I feel my hips moving correctly with my stroke.

Also, once comfortable swimming on one side, the drill progresses to rolling to center at midlength of pool, then stroke to other side, and finish length on next side.

The next drill has you count to 3 one thousand on side, roll to center and count 3 one thousand, then stroke to other side and count to three 1000.  Once I started doing these 3 drills comfortably, I really felt the transfer as I swim.

I did a TI weekend workshop some years back

I wouldn't say I'm any faster but I am much more relaxed, conserve much more energy and can swim a lot further at my steady state pace

I am no expert, and haven't taken the class, but I think the idea is to turn your whole core, so your whole body is angled a little. 

I think the goal is to reduce your resistance and to find a place where you "float" or glide. The idea is that if you can find your buoyancy and glide easier, you will be conserving energy. 

Body roll is more about using your back and core to power your stroke than about reducing drag.  View some Youtube slomo videos of great swimmers and you'll see that they roll about 45 degrees during the pull.  You'll also notice that they have skinny-ish arms, but HUGE lats (the lats are the motor, the arms are just the oar).

Technique work will be less efficient AT FIRST.  Typically it pays to work on individual aspects of improvement (body roll, head position, high elbow, etc.).  Over time you will see significant gains if you stick with it.  In swimming form trumps fitness.

The foundation of Total Immersion is very sound and has been validated in swimming community without a doubt.

Anytime you attempt to master a new skill, you will not be economical at first, will feel odd and will take more energy. If you stick with it, practice as instructed in books and videos, the progress will come. You will begin to feel it effortless at one point, you will know you are doing it correct as you will need less air, will be faster slipping through the water and will use a lot less effort. It ill come to you if you let it. Practice frequently and patience.