I would say you're allergic.

When I had Colby years ago, I was so deathly allergic to him, if I handled him at ALL, my eyes would literally swell shut for 3 days. I gave him to a friend who loved him and would care for him as he was meant to be cared for, but I visited him as often as I could.

I remember not being able to open my eyes all that well, due to massive quantities of eye goop. Though your reaction doesn't sound as severe as mine was, I would say you're allergic.