holyfuckballs
I deeply admire your honesty and trust. Welcome to the Thunderdome. We have some expectations to readjust. I've recently gone through it, so I'll gladly help.
PM me, we can sort this out for you.
Short answer: Yes you need a Flying Class 1 (FC1) and a FCC radio operators permit. Schedule a flying class one with a civilian AME in your local area ASAP to find out if you've got any hang-ups. If you have major medical issues such as PTSD, Heart murmer, etc...I don't recommend trying to hide...other opinions may differ. If you're comfortable and it's major, bring it up with the AME. If it requires a waiver, expect 8-12 months to get an waivered FAA FC1...so extra time is good. The FCC permit can be completely done online. Google it.
Longer answer: Congratulations, you are a USAF pilot. That entitles you to be...a USAF pilot. If you're a Colonel or higher, I highly recommend re-learning how to do things for yourself again. I say that only half-joking, as I've seen how those ranks get treated. In the civilian world: no-one cares who you were.
FAA has a COMPLETELY different grid of evaluating pilots. Respect that fact, as they are the ones who issue the license. Example: My instructor for the next week had over 10,000 flying hours BEFORE he was 26 and then hired by my current employer. Think about that for a minute. You're in a whole new world.
You are now a UPT graduate again who needs to figure out all the new rules of the road. At core, yes, you've got what it takes. The USAF and the FAA do not like each other. Get over it right now. FAA says you need FCC Radio permit. Go get it. (seriously it took 2 minutes of googling and something like 65 bucks, online don't be that guy) FAA says you need FC1. Go get it. The worst thing you can do is foster the mental habit of "but in the air force I..."
Just don't.
The background you came from has remarkably thin impact on your aviation future beyond how well it prepared you to be evaluated on your ability to fly airplanes...which will rapidly become self evident if you get hired.
Seriously. PM me. I've current on the hiring world right now. We'll get you sorted.