For whatever reason, x Nazi is not as offensive as you'd expect in the US. There was a recurring character on Seinfeld known as the "soup Nazi" and nobody got bent out of shape.
That said, substituting the phrase "The Grammar Police" doesn't even put you at risk and may be more accurate (since the complaint is that the person is officiously interfering in the affairs of others, not that he or she is a racist genocidal maniac).
MT_Head's answer is spot on — saying "I'm full" isn't rude.
I don't think there is another single word that is similarly polite and well-understood.
If you want to avoid saying "I'm full", you could say things like, "I've had plenty," or "I've had too much already."
Host: "Would you like any more?"
Guest: "Oh, no thank you, I've had
plenty. This was all wonderful, thank
you."
Or...
Host: "Do you care for seconds?"
Guest: (leaning back) "Oh, no I can't,
I've had too much already. Thank you,
it was delicious!"
Or...
Host: "Please help yourself, there's
plenty more."
Guest: "Oh, no, don't tempt me!"
( in a joking / laughing manner )
"No, I better not, thank you. This was all fantastic!"
All of these are ways to avoid saying "I'm full" which would sound natural. However, be prepared that "I'm full" is probably the keyword your host is looking for, so if they insist...
Host: "Are you sure? There's more dessert!"
Guest: "No, no I really can't. I'm perfectly satisfied right now, if I ate any more it would be too much."
In this way you can imply to your American host that you are not turning down the food because it isn't good, but because you really have eaten all you would like to eat — or even a few bites too much.
Really I think what this (sadly) reflects is that we Americans are prone to eating too much food when we really like it, so if you don't seem to have eaten "a little too much" your host may wonder if the meal wasn't to your liking.
So, while you don't have to say it yourself, it's probably best to understand that this is reason people in America say "I'm full." You're indicating to the host that the food was so irresistible you already had a little bit too much, and now you really must decline (or risk being sick!).
As an American this isn't part of my culture that I'm especially proud of, but it is what it is, and it's better to understand than to remain uncomfortable with it.
I hope this helps! Good question, by the way :)
Best Answer
As @Araucaria says in the comments, Grammar Police is an excellent alternative that conveys the fascistic tendencies of a police state without the genocidal implications.