When I get my orders can I book my own flight?

Contact your transportation office.

When I was transitioning to be stationed in Germany back in 2015, I was given a very specific timeline at Goodfellow AFB (AIT) to ensure all my transportation needs were met. Your PSG should set you up with an appointment at the Transportation Office/SATO to make the bookings. If not, now would be the right time to ask (start with your first line supervisor).
Caveat: If you are authorized to travel home before going overseas (I assume they would, not guaranteed), you may need to book a flight (or some other form of travel) from your home to wherever your flight will be taking you to your OCONUS location. Be VERY transparent with the booking office, and your first line, if this option is on the table.

Short answer: No. As CSM Darieus ZaGara said, contact your installation's transportation office - specifically your nearest CWTSatoTravel office (https://www.cwtsatotravel.com/searchResults.html).

If you're traveling on Permanent Duty Travel, then you're required to use the travel office for air, train, ship or bus travel. However, make sure you don't book tickets through CWTSato and skip the transportation office, even if you don't think you have any household goods to ship (it might be one of those "Ok, we have nothing to add" appointments or it might be "you didn't think of this and we'll have to make sure it happens for you" things).

I think since the military stopped shipping folks overseas on a troop ship, you fly over. I remember it started during RVN, I went over on the USNS John Pope, but later replacements came by plane and I went home by plane. AMC has contracts with a couple of charter airlines to fly rotators from BWI and Sea tack to OCONUS destinations. You would be Space R, not Space A as I would fly, so I not know how to register and roll call, etc.