The warning on a label of wine that says it "contains sulfites" simply means it contains sulfer dioxide. And as gross and scary as that sounds, it's a naturally occurring chemical compound found in all wines to some extent, even organic ones. Additional sulfer dioxide is sometimes added as a preservative, and in these cases, there are certain rules about organic wine and how much sulfer dioxide it can contain.
The other place sulfites come into play is in the sanitization of wine and winemaking equipment. Wine is a food product, which means there are health and safety regulations. But soap or other cleaning materials can mess with the wine. Better to sanitize and preserve wine with sulfites, which are already naturally present.
This isn't to say additional sulfite use is undetectable. Overuse can create aromas of matchsticks, burning rubber or mothballs not otherwise found in the wine. A girl in my class was particularly sensitive to this (also the scent of pear. She smelled that shit in every. Single. Wine. Except for the ones that actually smelled like pear). Once Shelby Ledgerwood figured out she was smelling mothballs in wines with more sulfites used, we all stopped hating that girl so much.
It's always good to be wary of what goes into your wine. Pesticides are certainly an issue, and one the wine industry has yet to figure out. Organic wine is different from wine made with organic grapes. Kosher wines no come in a variety of delicious styles, not just Manischewitz. Figure out what's right for you and go for it. But don't be scared of sulfites.
Also don't pretend you're allergic to them. Shelby Ledgerwood says true sulfite allergies are incredibly rare. It's more likely your headache is coming from tannins. More on those soon...
"Back in sulfite-ette cit-ay!" ugh the worst