Book your appointment before your draft is finished. This way you know you can get in by the time you have your finished product. You can book an appointment for a future date-- not necessarily just the same week.

Use Google Docs. When it comes time for your appointment, share your doc with Dr. C., but make sure that your name is in the title and remember to "grant access" via sharing.

Don't worry about whether your draft is too long or too short. This is part of the work you can get assistance with in the Writing Center; however, coming with a completed draft, even if imperfect, will be much more productive than an incomplete draft.