US university deadlines are firm and you need to ensure that the full application pack has reached the university before the required date. Some universities will allow students to start in January instead of August. Consult the prospectus for the January application deadline. Many financial aid packages are distributed in August so there may be less chance of aid for January students. Some courses, particularly MBA programmes, have rolling admissions, which means that applications will be accepted until spaces are full.
Early application is still advisable.
A priority date for admissions indicates that priority will be given to those submitting applications by that date, but later applications will be considered if space is available.