Tauriel, you need to pack everything up. If you drop the house and things aren't packed, they will fall to the ground (and anyone can grab them). If you do not get the choice to drop the house, check the house's moving crate (found under the house sign menu).
The moving trunk that Malakite mentioned can help you get packed up. Be sure to read about its commands under the commands tab of the website. Basically the process is to summon the trunk, load it, dismiss it, drop the house, place new house, summon trunk and unload it...being sure to lock everything down when you unload it. Remember, the moving trunk only sticks around for 2 days, then *poofs*, empty or not!
Ghost