The FAFSA form is used to calculate eligibility for different types of federal aid like Pell Grants, Work Study, and subsidized loans. It does not actually “give you money” and eligibility does not depend on how early you submit the form. Some individual schools use the form to determine how much you and your family will pay and the financial aid comes from them in the form of a grant (institutional aid) and that may or not be first come, first served but it depends on the college. Many private colleges use the CSS Profile to calculate what they expect your family to pay.
Your best bet is to read through the financial aid info on the websites of each college you are considering so you know what forms they want you to complete, the deadlines, merit scholarships that are available and if they have an extra essay or interview. Specific questions about the policies at each school should be directed toward their financial aid offices.