When changing houses, as stated, you get no cash back for the house
You do, however, get back any advanced rent you have paid.
So if you paid for the next 4 weeks you would get that back
To change house
Go into the Housing Panel on your character journal, and select Abandon
You can clear out the chest if you wish and collect any decorations. Assuming you have bag space to do so
If not, any and all items left in the House will go into the Escrow Account ... The NPC you interacted with previously, but got no response
Any items that are bound to other characters will be returned to them
IE If you let friends or kin members decorate your house.
Anything bound to them will go back to them
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