Pencil & paper encryption, some thoughts on polyalphabetic techniques
What you seem to be describing is similar to a single-letter book cipher, but using the solution to a crossword puzzle instead of a book as the reference source.
While this might have some merits as an ad hoc hand cipher — after all, book ciphers have also been successfully used as such in the past — it also shares most of the drawbacks of book ciphers.  In particular:

It has no secret key, other than the choice of which crossword puzzles to use.  Anyone who can guess that, and has access to those puzzles, can decrypt the messages.  This is a violation of Kerckhoff's principle.

It cannot encrypt letters that don't occur in the reference crossword.  No Q or Z in today's puzzle?  I guess you'd better not use those letters in your message.

Its security is strongly dependent on the encryptor's skill at choosing letters uniformly at random from the reference source — something that people tend to be bad at, especially if they're tired or in a hurry or just feeling lazy.  In particular, if the encryptor keeps reusing the same coordinates for repeated letters, or tends to choose nearby coordinates for successive plaintext letters that happen to occur close to each other in the puzzle (e.g. in the same word!), or even just generally favors a particular corner of the puzzle, the security of the scheme may be substantially weakened.


Some of those problems could be remedied; for example, superencrypting the ciphertext with a simple substitution cipher with a secret key (or, perhaps more simply, just using the secret key to somehow scramble the row and column labels) could make this scheme significantly harder to crack.  But other weaknesses, like the reliance on operator diligence, are harder to patch.