Almost certainly, sulfites are not causing you to have "red-wine headaches."
I made that comment briefly in this Los Angeles Times story this week, but didn't have the space to elaborate, so I will do so here.
The reason is simple. If you are allergic to sulfites, you wouldn't call them "red-wine headaches." White wines generally have much higher levels of sulfites than reds. There's a technical reason for this: reds have tannins to protect them from prematurely losing their fresh fruit flavors, while whites don't.
So if you can drink white wine, sulfites aren't your problem. You can test this theory by eating some dried fruit -- that has higher levels of sulfites than wine.
It's important to make this point up front because so many people mistakenly believe sulfites are some sort of evil chemical addition to wine, when in fact they naturally occur in grapes, and are crucial in keeping wine good-tasting and long-lasting for the more than 99% of us who are not allergic to sulfites. Some people are allergic to sulfites, and that's why the U.S. government requires wine labels to show "added sulfites." But sulfites are not the cause of headaches that occur only after drinking red wine.
So what is?
Every time I have interviewed doctors on this question, they tell me that people ignore the first and most obvious answer: Alcohol. Red wines are generally higher in alcohol than whites, often significantly so. Many people can't process alcohol well.
Can you drink a double shot of straight whiskey without a headache? If not, look no further.
If alcohol is your problem, the obvious solution is to drink red wines that are lower in it. That's a challenge in this era of big and bold wines, but it's not impossible. Try Beaujolais, Austrian reds, reds from the Loire Valley. Look for the alcohol percentage on the label and ask about it at restaurants before ordering.
For a cheap alternative, consider the Charles Shaw "Two Buck Chuck" wines. They're reduced down to 12.5% alcohol, which is pretty low for reds these days.
What if alcohol isn't the problem, and sulfites aren't the problem? Then what?
Unfortunately, then it gets complicated. Red wines soak on the grape skins for days or even weeks, and a variety of natural chemical compounds enter the wine in far higher quantities than in whites.
Check your sensitivity to tannins first, mainly because that's an easy allergy to work around. Black tea has tannin, particularly if you let it steep for a while. If that gives you a headache, you've got your answer.
While all red wines contain tannins, some contain far less than others. Grenache (Garnacha in Spain) is notably low in tannin; 100% Garnacha wines from Spain are often delightful. Other low-tannin reds include Gamay (Beaujolais again), Dolcetto and Barbera.
Avoid anything that says it was fermented or aged in new oak, which contains tannins of its own.
Merlot and Pinot Noir aren't naturally high in tannins, but you have to be careful to avoid the new-oak treatment.
Sulfites, alcohol and tannin are the three easiest problems to test for. What next?
Red wine is full of naturally occurring histamines (thanks to Jon Bjork for pointing this out in a comment). You might try taking an antihistamine before drinking red wine. That's a rather extreme solution, because antihistamines combined with alcohol can make you drowsy. Perhaps if you have other allergies acting up and are taking antihistamines anyway, you might try a glass of red that night.
Suppose histamines aren't your problem. Then what?
I wish I had a good answer for you, but I don't. Red wines are full of chemical compounds, and while corporate wines have a few added ones (like "mega-purple," a grape-based dye that makes wines darker), the overwhelming majority are naturally occurring. None of these chemicals are required to be listed in material anywhere, so even if you could figure out what compound bothers you, you can't know which wines contain it.
This is why some people claim they can drink one type of red wine and no others. Wine lovers sometimes say that well-made reds won't cause headaches, but there's no evidence of this, and I've never heard a doctor say it (and I've asked). Basically, if you find you can drink one brand of wine but nothing else, consider yourself lucky and keep drinking it.
Before giving up, though, I recommend trying biodynamic wines. Not "organic wines," which have no added sulfites.
Brief explanation: "Organically grown grapes" are a good thing. "Organic wine," in the U.S., is not. Without sulfites, wine quickly oxidizes and bacterial contaminants can make it taste nasty within months. The European Union allows sulfites to be added to "organic wine," but the U.S. government listened too much to some self-righteous, self-serving marketers of bad wine and wrote a bad rule forbidding this.
Thus most "organic wines" on the U.S. market are indifferently made wines from lesser growing regions that exist to serve a self-righteous but under-informed market (including, but not limited to, white people in dreadlocks.). If you like the aroma of fermented sock sweat, go ahead and enjoy them. Just don't kid yourself that they're better in any way. You have been successfully marketed to.
Biodynamics is as much superstition as science. That said, I've had enough readers tell me they can drink these wines without headaches, but not others, that I have to listen to them. I must point out that biodynamic wines tend to be lower in alcohol than others, and often don't have new-oak aging, so the answer might be contained above. But what the heck, just because a placebo is a placebo doesn't mean it doesn't work.
To summarize, before giving up on red wine, try testing your sensitivity to alcohol, then tannin. You might try taking an antihistamine. Then try biodynamic wines. But if you can drink whites, don't blame sulfites. And be happy that you still have half the world's wines available to you.
I'm sorry if this post doesn't solve your problem. Let me tell you something that might make you feel better. All my life, I have been allergic to tomatoes. I have had Italians ask me, "How do you live like that?" Quite happily. I don't mourn bolognese any more than a middle-class person should moan about not having Romanee-Conti every night. Instead, I just order pesto.