From [1], "Adjectives modify nouns; adverbs modify verbs, ..." - so though you're right that both sentences are grammatical, I think a lot of styles would have a problem with "a proper written sentence" since what you actually mean to say is "a properly written sentence" (unless you make it clear that you speak of "proper sentences", probably).
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"properly written"
Apologies for posting a completely off-topic comment, but as an ESL person this interests me.
I would have thought that both were different, but grammatically correct, with almost identical meaning.
"A properly written sentence" is a sentence that is properly written, and "a proper written sentence" is a written sentence that is proper.
But now I have achieved semantic saturation, and am not sure anymore.
From [1], "Adjectives modify nouns; adverbs modify verbs, ..." - so though you're right that both sentences are grammatical, I think a lot of styles would have a problem with "a proper written sentence" since what you actually mean to say is "a properly written sentence" (unless you make it clear that you speak of "proper sentences", probably).
[1] https://owl.purdue.edu/owl/general_writing/grammar/how_to_us...
Perhaps.
The way I read it, the choice of "proper written sentence" gives it the flavour of "Proper. Written. Sentence" - as in, with an annoyed emphasis.
As always, you can apply rules to English if you like, but sometimes they are optional.