Who do we address the cover letter to, and to where?
Also, is there any priority given to applicants who submit the application earlier?
The cover letter does not need to be addressed to any specific individual. Your resume and other materials may be reviewed by various offices and/or hiring officials.
You can address you cover letter to the U.S. Department of State, using the main Department of State address:
2201 C Street NW
Washington D.C. 20520
You can use "To Whom It May Concern" for your salutation.