I'd never really thought about it before but since coming here I have seen that this is a fairly common allergy. You might want to try a nostick pad wrapped with Coban, it's a sort of wrap that sticks to itself but not to your skin. It's also available as a generic.
I have allergies to the adhesives in bandaids also. I had a bandaid on for several hours 1 1/2 weeks ago and I still have part of the rash it left behind. For minor cuts, I can currently tolerate the liquid bandages, but they aren't nearly as good as the liquid glue treatments you can get at the doctors' office. For me, the 3-M brand used to be OK, but recently I found I can't use them either. I am planning on trying the paper bandages next time. They were recommended by my pharmacist when I asked last week. My doctor just told me to avoid bandages if I can.
I would love to hear if you find a solution that works for you.