Yes, that makes a lot of sense. I wish my other teachers had taught us about this when I studied Russian and French. I am sure they all hold their mouths and jaws differently!
I actually felt like Swedish forced me to be a slightly different person than I am normally. It's hard to describe, but to really get the language right it can feel like you're exaggerating emotions whereas I like to "play it cool" in English and I have a deadpan sense of humor. In Swedish I felt like I had to be more expansive and maybe even sound a bit more confident than I am.
Another thing was that it sometimes made it easier to say things correctly if I imitated someone I spent a lot of time with, using their intonations and rhythms. But then I would be confused as to whether I was copying their Swedish or actually "borrowing" their personality a bit, which was an uncomfortable idea!
RE: You can't speak "whale", with your "english-speaking" face"