Sulfites are not dangerous or scary, this is still a myth in the wine world. Sulfites (sulphur dioxide, SO2) are natures antioxidant and are naturally produced when yeasts ferment. All bread has them, grapes have them, dried fruits have them, anytime you cut into an apple you find them, even your own stomach produces them when it breaks down food. Anything that rots produces them. Those little juice boxes that kids drink have them as well.
The Romans and the Greeks used to burn sulphur candles inside their empty wine vessels to cleanse and sterilize them. This helped to keep the newly vinted wines fresh and stable. Winemakers these days, in addition to naturally occurring sulfite compounds, use sulphur dioxide usually in gas form to prevent microbial growth so the wine will last longer and not just turn to vinegar. It can be applied to just picked grapes so they don’t prematurely rot. During the crush faze it can be used to keep ambient yeasts from burning off before commercial yeasts are thrown in. They can be added to stop the fermentation process or to avoid malolactic fermentation. It is also used during bottling to help preserve the juice. If a winery is spotless and germ free and the resulting wines are consumed young the extra addition of sulfites are not completely necessary.
Sulfites do not cause headaches. If you are severely asthmatic and have a severe allergic reaction to a sulphur dose 100 times more than what you find in wine then you will develop respiratory distress. This affects less than 1 percent of the population. Headaches are most likely caused by dirty equipment used in the manufacturing of mass-produced wines, or perhaps over consumption. Have food and water with wine, this will help with the headaches, plus you will enjoy it more.