Since FAFSA stands for Free Application for Federal Student Aid, the name already indicates that it's financial aid on the Federal level. This means that any college out of state will accept this form of financial aid. Deadlines vary depending on the state, but you can check the state's deadline on https://studentaid.gov/h/apply-for-aid/fafsa (scroll a bit down past "FAFSA Announcements" & there should be a section called "FAFSA Deadlines.")
So let's say I'm from New York State & I applied to a school in Massachusetts (ie: Boston University). Although BU is an out-of-state school for me, I can fill out my FAFSA & once it's processed, it will show me the amount of student aid I'll be granted.
I hope this answers your question! Please let me know if I need to clarify anything.
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Ditto to that, it is better to get it done and over with early so you don't have to worry about missing deadlines.