You live where ever you want, as long as it's in the Continental United States. They buy you an airline ticket from your home to your TDY assignment. At or near the end of that month of TDY, or the beginning of the next one, they buy you an airline ticket from wherever you are, to your next TDY location. Repeat this every month.

Last I knew, the official position on you going home while you are in the homeless base was going to be your problem. They are not going to help you go back and forth.

Also, your ability to take a weeks vacation while in the homeless base is somewhat questionable. Currently, if a pilot from a regular base has vacation during a month they should have been on TDY, they skip that person and move to the next. It's too hard for them to figure it out. Not sure how they will handle it for people on permanent TDY.