I wonder if 'play cute' or 'play adorable' is frequently used to stand for 'act cute/adorable' in spoken language. It seems easier to google out 'act cute/adorable' instead of 'play cute/adorable'.
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These have different shades of meaning. One who "plays cute" is intentionally giving that impression, while one who "acts cute" may be doing it intentionally but may not.
I was always under the impression that "don't play cute" means "don't pretend to be innocent"/"I know you're being dishonest".