A high EFC doesn't preclude you from loans. It precludes you from need-based scholarships and grants, but you can still take out federal loans.
Because I haven't had coffee yet, I'm blanking on how Staffords relate to EFC, but I know for PLUS the EFC doesn't really matter. It's unsubsidized, so it's more like a personal loan that the government happens to guarantee.
If you file your FAFSA online now, you should be able to move forward with your loan applications (also online) immediately. If you have endorsers, they can fill out their bit online as soon as you finish. It'll take a day or so for it to get to the uni for certification, but it will leave you with an only slightly delayed disbursement.