I share your pain. I've been getting more and more sensitive over the last 6 months or so. Right now I can wear them for an hour or so, but any longer and my ears start to swell fast. Mostly people are suggesting to just get earrings made out of the hypoallergenic metals, like titanium or
argentium silver. What I'm doing now is coating all my earrings with nail polish and I've also ordered some coconut oil, apparently if you put some on the wires it buffers the allergic reaction. But for long term it's probably best to change out the wires for ones made out of surgical steel or something like that. Your reaction sounds pretty bad so you probably shouldn't risk it and just change out the wires for better metals or buy new earrings.
I asked this same question on reddit a few days ago. Got some helpful answers. They directed me to this article. It kind of sums up the options, most importantly it mentions the diet. I'll see if I can manage my sensitivity with a low-nickel diet. Apparently, grains, beans and seedy foods have lots of nickel. Try eating green vegetables. Also interestingly whole grain foods contain more nickel than the processed versions I eat whole grain pasta and bread almost every day which kind of explains why I'm getting this nickel allergy suddenly. Anyway, good luck!