Be careful of packing your own belongings (PBO) - I have been warned by many that this is a "big neon sign" to US Customs as a more likely target for inspection!
Although this came from removals companies, they did back this up - The removals companies have a lot to lose (their well-earned reputation with customs officials) if they mis-report any inventories, or ship items that shouldn't be shipped.
Nevertheless, to update you on our own quest for quotations for shipping from Southampton, UK to San Antonio, Texas...
Although we haven't chased any of the companies, only three or four have bothered to reply!
We have a two bedroom house and are taking very little furniture.
We have had two of the above companies come out and assess what we are taking, both of which have estimated the volume to be 6.3 cubic meters (230-odd cubic feet).
The quotations we have received have ranged from a little over £1600 to £1800 to as much as £2300, so shop about.
The cheapest option is not always the best to take, but for us, it is likely that we will be choosing the cheapest, as the company are quite local to us here in Southampton.
Not wanting to advertise or promote anyone, Leatherbarrows have been the most helpful so far, with F+N, AGS and Bargate Removals being others who had actually bothered to reply to us.