The Art of Listening
One of the most effective ways to catch errors and awkward language is to read your writing aloud, particularly if you can read it to someone else. It's not necessary for them to comment if you don't want them to, but it is a good way of forcing you to read slowly and listen for awkward words and phrases, missing punctuation, and information gaps.
If you're on a deadline, try to finish a final draft early enough to leave yourself a few minutes to read the piece through. Working on a long research paper or a book-length project? Try reading the material aloud in chunks over time, and mark corrected sections as you go so you don't duplicate your efforts.
This is a surefire way to catch critical errors before editors and readers. And just a reminder: Once you've made the first set of corrections, be sure to read it again all the way through.