We did this move two years ago. We shipped our stuff by shared load because it was more economical and we weren't in a hurry.
We found the company through an expat forum on FB and doing as much research online about the companies.
From our experience get a company that can deal with the import and duties that might be applicable. Our boxes were opened and resealed by customs so make sure you boxes properly list the contents. Also make sure you get strong boxes. We had some from a moving company and some from a chino. The chino boxes barely survived, the others were great.
We've also done the UK to Portugal and UK to Spain. We did it pre Brexit so moving was easy, however, the same thing applies, sturdy boxes and a reliable company.