I moved from NH to CA (and back). I used a U-haul, but one that was towed and only went one-way. I found it easier than shipping, plus I didn't have to worry about my items geting lost. I had some family furniture, my kids baby stuff - things that were not replacable if lost.
The U-haul wasn't all that expensive since I used one that you tow instead of a full U-haul truck. The gas cost around $600 but gas was much more expensive at the time I moved. (I won $800 at a casino in N.M. on the way through, so that covered gas and hotel stays.) It's between the two of you how you manage the money portion. I was in a better financial situation than he was, so I would have paid for it all, including his flight out to me, to make it easier. We just lucked out at the casino to cover what I had already planned to pay.