I moved from Scotland to Finland and also shipped the majority of my stuff via a container on a boat.
Of course I downsized considerably before moving, leaving behind all the obvious bulky things that made no economic sense to transport such as furniture, electronics, etc.
In the end I think I paid to ship 8 cubic-meters of "stuff". It was a couple of pallets which were around 2mx2mx2m when combined. Things arrived smoothly but I recall quite strongly that I paid more for the insurance than for the actual shipping. Happily I didn't need it.
I'm not sure it would make sense for moves within the US, since everything would probably go by truck anyway.