I’m pretty sure white and red wine both have sulfites, I don’t see why the quanitity would be different depending on the color. If white doesn’t cause a headache, I think you know it is not the sulfites, unless the white was specifically aged for a long time in oak barrels, which would leach tannins also.
White would have less tannins (unless it is aged in oak barrels, but still less than red, because the grapes used for red wine have more tannins than the grapes used for white) but still has some anyway, so if it is the tannins causing your headaches, white should affect you less, but I still would not overdo it.
Tannins are found in large quantities in black tea, walnuts, purple grape juice, red cherries, and many berries, and I am sure there must be others.
I know very little about sulfites, but another jelly can probably answer it or a google search.
Or, just give up the wine, take the extra money you now have, and treat yourself to a nice vacation or two.