Can I move my family to Bragg on post while I am in Korea?

Why didn't you ask your Career Counselor? This is exactly what's supposed to be covered in a post reenlistment counseling.
Yes you can move your family to your HAAP. Most likely, Port call will only provide you with a ticket to your new assignment and only within a small window of time. However, you can go to travel and have it amended. You will pay the difference in price between your old ticket and the new one.
As to getting your family on post, that will only occur if your HAAP is in your orders. Specifically, your MPD or levy section will cover this in their levy brief.

I'm not sure how that works. I'd recommend calling Fort Bragg Post Housing office at: [login to see] , or on line at: http://bragg.corviasmilitaryliving.com/

Personally I prefer to live off post.

A big chunk of your answers are in AR 55-46, chapter 2. Mainly talking about the designated locations. Additionally, if you are going to assist with the relocation of your dependents while PCS, make sure you have that statement is added to your orders. You will also want to peek at AR 614-200, Chapter 9. And the other piece is you will need to make contact with the gaining installations to see if they will allow you to move your family "onto" the installation prior to have actual orders to that installation. Some will require orders assigning you that installation before allowing you housing and other things. The regualtion states that HAAP's can be changed, do your research..