The folks who do and read lots of comp research tell me that several things help students write better:
1) Writing lots and getting feedback
2) Writing for different purposes, talking about purposes, getting feedback
I've taken that information to heart. The assignments my students have turned in so far have all been short, low stakes assignments. In two of the four cases, the assignment asks them to write a paragraph. A single paragraph of writing gives me a chance to underline something and write in the margin "interesting idea" or "indeed!" I can give short responses, mostly responding to an idea. And I can mark one grammatical problem (if there is one), or remind the student to cite the text, and because I've marked one thing and it's there along with a positive comment, the student can manage looking at it and thinking about it. (And the grading isn't overwhelming for me, either.)
The paragraphs are also a starting point for one bit of class discussion. So those students who feel slow-tongued are prepared and can use it to say something.
One of the four assignments is a summary of a short reading, also short and low stakes. But summarizing is a skill students need to practice, as is reading carefully, so this written assignment is a start.
The final of the four is a short diagnostic piece (for the writing class) about why they're here at NWU. It's ungraded, and I use it to get to know students a bit, to spot really big writing issues early, and to have a chance to give students some positive feedback on their writing.
In each of these courses, students will write other sorts of writing, and will continue to write short pieces as well.