What a great question!
While I don't think that there is any One Method to Rule Them All, I do notice that many students can articulate themselves well in conversation, but struggle to transfer the same depth and nuance to their writing, and I think that has to do with the way writing is often taught in Common Core.
As a writer, editor, and tutor, I've found that there's a lot of fear of "doing it right" or a feeling of permanence that comes with writing that isn't there in conversation, so the first step is simply to give students permission to revise and make revision a common practice. I've also found that practicing the stream of consciousness method of writing helps build the muscle for transferring depth and nuance from the mind to the page--just ten or fifteen minutes of free-writing per day with the ability to share if they'd like but also to keep it private if they'd prefer can be incredibly helpful. While the student is building those practices individually, I often scaffold their writing during sessions with questions as we go--and I encourage them to write their answers down right after they tell me.
I also think the lack of focus on grammar and mechanics is a major issue in Common Core. The more students understand the parts and inner workings of a sentence, the more likely they'll be to know how to write strong sentences or fix weak ones on their own. I highly encourage the practice of sentence diagramming as well as building vocabulary with Greek and Latin root words to help them identify those pesky SAT/ACT vocab words and strengthen their sense of how languages interact with one another. To this end, I highly recommend Grammar Girl for adults and young students alike, as she has a phenomenal talent for making the grammar fun and understandable, and for teachers and advanced students, I also recommend reading books about copy editing such as Dreyer's English by the copy chief of Random House (where I work), which also has a fun grammar game that came out this summer called Stet!
For teachers or students who are more advanced, almost any master writer you can think of has at some point in their career written a book on their writing craft, and I encourage those who teach or would like to learn at a higher level to identify their favorite well-known authors and read those "on writing" craft books. I also highly encourage teachers of writing to take not just professional development in pedagogy, but in writing--either a workshop or a masterclass or a webinar can be incredibly useful. Literary magazine A Public Space hosts great master classes taught by contemporary writers and editors, She Writes and Masterclass have great webinars, and there are plenty of conferences and other professional development opportunities available.