CW is CW and digital is digital. They are not interchangeable. Unless it is a contest like the IARU World Championships, The School Club Roundup (which really isn't a contest I guess), Field Day (which also is technically not a contest either) or one of the state QSO parties, contests are usually mode-specific.
There are a few PSK contests, but RTTY is by far the most popular mode for contesting.
Here is a link to one of the best contest calendars around: http://www.hornucopia.com/contestcal/weeklycont.php
You can go to the 12-month or Perpetual calendar tabs and get an idea of when the contests are scheduled.