In November 2001 I shipped a small house worth of things, mostly clothes, books, records, CDs, kitchen utensils, plus three beds, two wardrobes, two chests of drawers, and a dining table and six chairs. In all it was about 930 cuft, just short of a 20' container full.
I shipped it with Allfreight International (
www.allfreight.co.uk , 01932 853344, e-mail: sales@allfreight.co.uk) and I can't recommend then highly enough - the sevice was excellent. They sent a team to pack, inventory and load the household contents, and everything was paid for through to delivery to an address of my chosing in New York. They're based west of London, so might not be suitable as you're in Leeds.
They also took care of the airfreighting of two trunks of clothes and basic kitchen utensils until the sea shiptment arrived.
I think we paid around £2,000 for the main surface shipment, and about £120 for the two trunks airfreighted. Plus insurance in both cases.
You are right to get several quotes as shipping companies set a price based on the resources they have available - you need to find a shipper with few bookings on the day you want to move.