Good post, Eric.
You might try, bite my tongue, using a kick board. Turning your head first to one side, then the other. Feel the differences. Try to work through them. Do this for a few laps every time you go swimming. It took me a couple of weeks of this to have it not feel totally awkward, but it did help a lot.
Another thing is to go back and spend time breathing on your traditional side, thinking about and analyzing all the movements and feelings you find there. Then try to transfer them to the new side.
Look for differences in the spearing angle of the lead arm and differences in the catch timing.