Hi Emily, it's almost certainly an allergy to the metal in your flute. I had terrible problems with this when I first started playing but realised it was linked to my allergy to any jewellery that isn't silver or gold.
The first flute I had to learn on ( a long time ago

) was a school Jupiter Sapphire which is has a nickel silver headjoint which is one of my worst for allergic reaction. I saved up for a Yamaha 311 which has a solid silver head and have not had any trouble since, even though I still have terrible contact dermatitis where my skin breaks out in exactly the same weepy sores you've described if I use the wrong soap, washing powder, cosmetics etc. Can you trial a flute with a solid silver head to see if it makes a difference?