First, it is total baloney that this is a conditioning issue -- I am overweight, unfit, and asthmatic, and I can swim 2 miles without getting out of breath.
Either you are holding your breath, or you are holding tension in your chest.
For #1 -- immediately after inhaling, exhale continuously but relatively gently. For a long time, I would hold my breath after the inhalation. Don't do that. On the other hand, don't force the air out too fast, or try to get more air out of your lungs than is comfortable.
For #2 -- don't try to "tone" your body, or "extend your bodyline" -- imagine your body floating like a lily pad on top of the water. To generate your inhalation, as you bring your recovering arm forward it will open up your ribcage automatically -- if you are relaxed. If you feel tension holding your ribs down (or your shoulder hunched up, or your arm not moving freely forward) relax that tension, until ... the movement of your recovering arm opens up your ribcage for the inhalation.
Then, go back to #1.