Oh no.. that's one hell of an allergic reaction! Stop eating eggs in any form and toss what you have. I only say this because there are people who are so allergic that even if egg touches them they go into anaphylactic shock and have to carry an Epi pen. You're reacting stronger and stronger so it's better to be safe than sorry until you can get confirmation on whats happening. I hate to suggest this but if you're opposed to seeing a doc and want to self-test try just a bite or two of egg whites. See if there is a reaction. Wait a few days and try just a few bites of cooked egg yolk. There are people who are allergic to one or the other. But be careful!
Serious reactions kind of freak me out so if I were you.. make an appt. with an allergist so they can run tests. Are you taking any allergy meds? Before you see the doc go off them for a few days before as they will affect the results. If you're on anything else make sure the doc's office knows so they can tell you if any of them will affect the tests as well. I was an allergy ridden kid, happily luckily grew out of 95% of them - I definitely feel for you..
I'm curious.. any trouble with potatoes, tomatoes, peppers, eggplant? Sometimes it seems that some people who react to those items that fall under the nightshade category also have egg trouble.. as well as trouble with chicken. It's just a weird cross-allergy..
Re: Paleo.. there are many people who are happily fed without eggs and they are definitely growing big and strong. Meat, fish, veg, dairy, fruit, good fats, etc. If you like the texture of eggs you can make a nice creamy cauliflower mash and top with your favourite omelet options and I think you might be happy :) Be well!