Re: "Don't use strength!"
"Have someone hold your arm, and just relax it and let it drop. See your fingers relax and your arm will become heavier. It seems to me that when someone's fingers are tense their body is tense also. "
If you're really entirely relaxed, though, your posture falls apart. And your arms drag behind you when you turn if they're genuinely totally 'relaxed'.
I think there's something everyone is trying to express by things like 'don't use strength' or 'relax' and people who know how it feels know what they mean, but those words aren't really accurate so if you don't know the feeling the words don't really convey it...
(Personally my experience with exhaustion tends towards sloppiness, too... not that practicing when exhausted isn't interesting for other reasons, but for me I don't find it makes my techniques better!).
Personally I sometimes find it helpful to think of being 'correct' rather than 'not using muscle'. I.e., it's not just the absence of using a lot of strength, it's the presence of some other principle. I can't always tell if I'm doing it right but thinking of it this way is for me often more helpful.