It does sound like anchor worms, but before I'd have you grab your fish (which may hurt them more than help while they're ill), I'd like you to look at this picture and confirm whether there's anything like this stuck to your fish.
Now, I'm actually really worried about your fish, because I'm rather sure that they haven't been in a fight, but gotten sick because of bad water quality. So tell me, do you know about the nitrogen cycle? You were talking about high nitrites, and that's very good that you checked
Did you test ammonia, nitrate and pH as well? These are the most important parameters, and you should always test the whole set whenever you see your fish acting strange
What worries me, is that your bio filter might not have been ready for the biological load that two goldfish put on the system, as in, your tank might not have been cycled yet before you put your new fish there. Especially since you mentioned "washing the tank out". You should never clean out the whole tank.
Please tell me how big it is! And something about your maintenance schedule: how often you're feeding/testing water/changing water. For how long did you have it?
Finally, if it IS anchor worm, you should follow Minde's instructions
First get rid of the worms, then add the anti-parasite medication to kill free swimmers, and add something for the fishes' wounds, like Melafix