The word brown is doing you a disservice, and once you replace it the lists stop contradicting each other.
What actually matters is carbon-to-nitrogen ratio, not colour. Greens are wet and nitrogen-rich; browns are dry and carbon-rich. Some brown things are neither — dried blood is brown and behaves entirely as a green.
So, your candidates:
Cardboard — yes, and it is the best thing you have. Plain corrugated is excellent: high carbon, and the fluting creates air pockets, which is your other problem. Tear it up rather than laying sheets in, or you get a waterproof layer. Ordinary printed boxes are fine; what you skip is glossy coated card and anything with plastic tape or heavy foil.
Newspaper — yes, with a caveat. Good carbon, but it mats into a dense wet layer if added in wads. Shred or tear it and mix it through rather than layering.
Sawdust — yes, but sparingly and only untreated. Very high carbon, so a little goes a long way, and it compacts badly in quantity. A few handfuls mixed through, not a bucketful. Never from treated or painted wood.
Other things you almost certainly have: egg boxes, kitchen roll tubes, paper bags, the inner tubes from wrapping paper, dry stems from anything you cut back in autumn, and untreated wood shavings from a pet.