Ah ok, we always called them fleece or fur bearing animals.
I wasn't sure you that were not talking about fiber 'stuffed' animals (pillows) for the corner of a room, or a bed decoration after you said you did not have room for real animals.
lol
We also have so many of those that we don't have room for any more.

It's not for fur. That requires butchering the animal to harvest the fur. It's the fiber or fleece you're harvesting. I've seen people just brush them to get fiber which has the advantage that it is not a all scratchy. I've seen yarn spun from everything from rabbits, dogs, goats, sheep, llamas, alpacas etc.
Yes, I'm not looking for animals who have useable pelts or skins with fur, but animals who need to be sheared for their fleece or combed for their hair.
In that case I would suggest sheep for a start. Im not sure but can't they be raised 'with' your goats, sharing the same space?
My sister has been talking with someone on her goat forum that also has sheep, and apparently it can be tricky to keep them both together. The biggest issue is their mineral requirements are very different, particularly with copper, which the goats need a lot of and the sheep shouldn't have at all, which means you have to figure out some way to allow the goats, but not the sheep, to access the free choice loose minerals. The other issue is that, in spite of being somewhat similar animals, they don't really speak the same language, especially as far as figuring out their pecking order goes. Goats will rear up on their hind legs to butt each other, but sheep will back up to get a running start and then "ram" each other, which can lead to the sheep not being able to defend themselves against the goats as well. It can work between certain goats and certain sheep, but some individuals just can't seem to get along well. That was my sister's friend's experience anyway.