The cat's paw tool is very good for non-spiral nails.
Get your circular saw, and set it to the 'thickness' of your deck boards. Set the saw to cut NEAR the nails; not too close. If you set your saw to the thickness of the deck boards, you don't have to worry about cutting the joists or anything. Do the cutting new BOTH ends of the boards you want to remove. Then, if you want, you can try to pry-up the small 'blocks' that remain nailed to the joists. A cat's paw or a small crow bar should work fine. (Just get UNDER the little bugger and pry up.)
Depending on what you have for deck boards, you may want to leave your new boards out in the 'open' for a short while to dry out. If you put them in while they're wet or something, they'll shring in place, thus, 'pulling' on your screws. (You will use screws with the NEW boards, eh?) WHen you go to the Home Center, tell the Hardware 'guy' what you have for boards, and for thickness, and they'll set you up with the right screws. Be sure to PRE-DRILL your new boards. (Coated screws work best for me.)