Hmmm good question. I'm also allergic to many products....

If the items are all cotton they don't need fabric softener in the first place.
You might try re-drying them in the dryer with a towel that wasn't in the problem loads. Since it was just a sheet and not liquid, it's more likely just on the surface of the cloth (to prevent static cling, which as I mentioned, cotton doesn't get.)

I'd have to put a sticky on the wall to remind me not to purchase said sheets again... or I'm subject to do it again. Snuggle is good for me too, but not the wildly scented ones. Fortunately, most of my clothes are cotton now, and I don't need them much. (Only a piece of one for my bed linens.)

You probably don't need to wash them over, but rinse water cycle only if the dryer idea doesn't work...

Good wishes... let me know what works, ok?