Good writing is rewriting. That said, in my experience the hardest part is writing the first draft, i.e., just getting something down at all. So do whatever you need to do to get your own thoughts on the page (except Claude, obviously), and then proofread, correct, and revise. If you do substantial revisions, then you'll have to do more editing passes. When you do a pass and only make minor changes, that's a sign that you're probably done.

If you think you're just a poor writer...well, you won't get better without doing it a lot, will you? Also consider that AI assistance will actively hamper your growth, and that's considered a feature, not a bug (would the AI labs prefer you to depend on their products, or to outgrow them?).

On spelling and grammar, you can't get flagged as AI for using spellcheck. I don't use grammar checking, but I read somewhere that grammar checking algorithms have gotten good now, so that could be something else to look into (as long as it just highlights potential problems and doesn't rewrite your words).