I ship my bike itself using shipbikes.com/aircaddy, but box won't accommodate a filled pannier.
The shipbikes aircaddy is pretty big, you could probably stuff empty panniers in easily and pack the gear in the empty spaces. Maybe use multiple plastic bags in sizes to fit the spaces.
Going cross country so need all four panniers.
Personally I never found that I needed more for going cost to coast than I do for any of my shorter trips. I have gone coast to coast with as much as 55# of gear and as little as 14# of gear. On the U/L tour I had full U/L backpacking gear (14#) and the bike (38#) all in my soft case (mailed it home at the start). It was all still easily within the airlines weight limit and it was easy to carry the whole thing into the airport unassisted. If I wanted to carry more a second bag would still have been free since I was flying with southwest.
So there is a wide range of approaches ranging from not even needing panniers (I have used a couple stuff sacks on the rear rack, or one in a bar roll and one on the back) to fully loaded with 4 panniers and stuff on the rack top. Then there are bikepacking bags which work especially well for lighter loads.
Having done the wide range of packing styles, I have found that I prefer the lighter end of the scale. It is really a joy to ride a sportier much lighter loaded bike. It may not be for everyone, but it is one option to consider.