I have celiac disease and (at least at this point, about 7 months into a gluten-free diet/healing) I'm so sensitive I have to avoid foods processed on machinery that had wheat products on it.

For seaweed, I'm not sure what this could include: maybe soy sauce, maybe other kinds of food products. When I question a food, I find celiac message boards really helpful. Here is a thread about gluten in nori that should help you find a brand that is safe.

In general, since it seems you're very sensitive, you might want to stick only to foods that say they don't use shared facilities. It sucks, but it's better than getting glutened.

We get them from the Korean marketplace and have no problems. Where are you purchasing them from?

I know that they do add oil to the seaweed sheets, so if the ingredient list doesn't have some kind of oil in it, there is a possibility that there are unmentioned additional ingredients.

Are you sensitive to even tiny amounts of gluten? There might be cross contamination at the packaging factory if they also handle gluten containing products. Have you tried a different company? No idea if this is the case for you, but I think it could be a possibility.

How much are you using? Seaweed is high in iodine which can muck up your thyroid and cause odd symptoms if you're sensitive.

I read that some seaweed/nori use fillers which are made of wheat

As far as I know, seaweed doesn't have gluten. So most likely cross contamination, or seasoning. Are you using any seasoning? Is the seaweed seasoned? What brand is it?