I'm glad you were able to see your Doctor! I hope you are doing better.
Now, just to figure out what you're having a reaction too!
Just thought here - I don't know where your located - do your chickens free range? Maybe they got into a plant (like Poison Oak/Ivy) and got it on their...
Did you treat the chickens?
Could be you are allergic to the chickens/their feathers/dander or the mites.
That looks like hives to me, if they persist or spread, then consulting your Doctor would be a good idea.
Do you see any bugs crawling?
It's possible that they could pick up lice/mites outside, look over the chicks - part the down and feathers and see if there are bugs.
What are you using for bedding? Possibly you are allergic to bedding, dust or even the chicks themselves? Are you washing your...