I haven't been in the situation where contraception made me ill, but I have chosen to stop putting those hormones/chemicals into my body. But if contraception was actually making me feel bad and giving me life altering illnesses (like migraines, pain and very heavy bleeding) I would absolutely stop putting that into my body. Clearly your body doesn't agree with it.
If eventually having more children isn't off the cards, could you maybe use the rhythm method (timing your cycle and having sex only when you're not ovulating) and then use condoms only those few days that you're fertile?
Otherwise, I would try some other sort of hormonal cont. and see how it reacts with your body, the reactive hormones or doses are all different, and you might react better with something else.