I would definately get my things moved before the 4th day ended, just to be safe!
I usually organize and move my keepsakes to my bankbox first. Equipment and supplies that would not be readily replaceable due to price or rarity are the next to be moved, along with anything from my bankbox that needs temporary relocation to make room for my collectibles...I try to divide these items up into groups once I've finished moving them to the new home. Everyday items and common deco/furny are usually next to go, followed by crafting resources and consumables.
If you have a Furny Dye Tub handy, it's easy to color-code your moving containers. This can really make finding things alot easier while you get settled in your new home and find time to unpack.
Don't forget the newest addition to 1st yr. Veteran Rewards - Commodity Deed Box, the homeowner's helper!
This item allows you to turn bulk resources into Commodity Deeds or Commodity Deeds into resources, right in the comfort of your home. Definately more convenient than pack horses/llamas or beetles!
If you have a Crafting character, especially one that's still training skills, you may want to choose this Comm. Box as one of your Rewards. If you're out of Vet Reward picks, you can purchase the Comm. Box from other players too.
Good Luck with your move, and I hope you get lots of enjoyment from your new home!
