You should be checking more than just your spelling. Grammarly catches syntax and grammar mistakes I make when I'm writing quickly. It also fights me on every other comma but hey, you win some you lose some.
There is no better way to work collaboratively on a written document. It's what we use at the day job for blog posts, website copy (pre-implementation), internal documents, and more. Pretty much anything written.
This is our go-to work management app at the day job. More powerful than many tools out there, but still simple enough to get the hang of after a couple weeks on the job. I personally prefer Notion, but Asana is pretty good too.