The only significance things hold is the significance we attach to them.
If the stuffed animals have significance to you beyond their utility value in helping you sleep better – however they do that – then that is their significance to you. It’s unlikely that a child, upon receiving your beat-up, drooled-upon, stained and shapeless stuffed animal would attach the same significance to it (assuming it was not so bereft of toys that this was a one-off, as in The Gods Must Be Crazy, where sacred significance is attached at a tribal level to a used Coke bottle).
So, yeah, you might find a child who would appreciate that thing “as is” ... but that’s not likely in the world that most of us inhabit. (That might be different among some family members, who may appreciate a gift ‘from you’ that contains whatever significance you have ascribed to it: if you offer it with a compelling story, then that could help.)
I would suggest throwing it out in the recycle bin as a useless-to-you used object of no particular significance (the fibers are generally recyclable, and certainly if the stuffing is some kind of natural fiber), and if it makes you feel better, donate a new toy to a worthy charity. Most kids do seem to attach significance to gifts that they receive when the gift is obviously new and objectively ‘valuable’ to anyone who can see it.