Personally, I would not use any paper and cardboard for brown composing. There are many chemicals (ink, stabilizers, protection to UV, UV fluorescence, etc.), which would undermine the main reason to make composting.
But also your greens, if it start to smell (greens from garden), there is something wrong.
In general green will generate compost also without brown stuffs. Just more slowly. But I think it is a general problem to all people: in summer there are too much green and on late fall too much browns. But nature work in such manner. Mixing helps for late spring, but I would keep some branches for summer: these branches give some N, but helps also to aerate the structure (in case of lawn green) and help to have better compost. On the other hand, in autumn you need less compost output, so if composing will be slower, let it be slow.
Try to identify why insects will reach your green compost and why it smell so much. In theory a good compost would attract many insect (a mix green and brown), and also larger animals.