You stabilize the wine to prevent re-fermentation with Potassium Sorbate. Sulfite's are added not as much to stabilize but to protect the wine from oxidation. Once the wine is stabilizedthe yeastwill not produce CO2 as they are now dormant which is heavier than O2 and provided an insulating blanket on the wine that shielded the O2.
Most are not truly allergic to the sulfite, they are highly sensitive. I, like you an sensitive as well. If I drink more than a couple glasses (one glass withmost redwines) I get a terrible headache and my nasal passages feel swollen. One of the reasons I prefer the kit and home made wines is the fact that our wines have considerably less sulfite's than a commercial winery wine. I can drink more and not suffer the side effect from sulfite sensitivity near as soon.