If you move to Great Britain, you gain relief from import duties for personal possessions. I imagine most other countries offer similar arrangements, and you should be able easily to find information on the individual country's government websites. I suspect that the simplest thing for duty purposes is to send stamps etc with the remainder of your household goods.
In this case "best" and "most economical" may be in conflict. Faced with the same, I would probably buy a premium airline ticket to assure overhead cabin space, pack the 6 or 7 most valuable binders into a roll-aboard bag, and take them onboard with me. They will be subject to customs inspection but could be below the value limit, or exempt as antiques. Those not fitting in a roll-aboard bag I would likely ship via FedEx International between two business addresses as it seems to be the most reliable and in this case I would value reliability over cost. If the move relates to a job change, the cost may be deductible as a moving expense on your 2021 US tax return.
I moved my stamps from Europe to the USA then to Thailand then to Singapore then to Thailand then to the USA, then to Thailand again.. Most of these moves were by sea shipment, packed as part of my household goods. Never any issue with customs in any country. I also hand carry in the airplane several times part of a collection. Shipping by sea. It can get very hot or cold and wet with high humidity. Pack each album in a strong plastic bag and seal as good as possible with tape and make sure the albums are well closed. You do not want the stamps to fall out because the box is upside down during the shipment. Shipping by air. I never liked the plane too much. As personal luggage there is always the chance that Customs wants to open it. If the stamps were in the luggage hold they are ice cold and when opened by customs there will be immediate condensation on the stamps ruining the mint stamps. Shipping as household goods by plane is quite OK, customs will not immediately inspect (if any) because the goods first will go to the clearing area waiting for the freight forwarder.