woman who reads too much
If you mean the type where the hay is used as a temporary structural wall: yes you can, they will just rot down a bit faster than unspoiled ones. If you mean the type where you spread it on the ground and plant into pockets of soil or potting mix, same story.
Just be cautious that what you are planting in them won't be affected by the same type of mould that affects the straw, and keep leaves trimmed off from lower branches if there is mould all the way through the straw so that it won't be transferred by water splash: this is much less of an issue for non-edible plants.
And wear a mask when playing with it, as lugging it about will send the spores into te air and your lungs, which no lungs enjoy.
Your house and garden photos always delight me, even the 'weeds'!