As sem613 and Best Bubby say.
When we moved from EY to the UK, we shipped almost all of our belongings by mail, including all of our seforim except for a handful.
I have trouble believing there is any cheaper way. I definitely recommend the mail. However as sem613 says, get the boxes from the post office and reinforce them with lots of tape. Use the extra strong brown tape for all corners and in several directions around it, and use see-through tape for all other areas such as the addresses. Also we put some stickers with the FROM and TO addresses on several sides, covered with see-through tape, again. Just to be sure nothing gets lost, you can write your phone number on it as well.