If you do find yourself in the company of North and South American Others (Canadians, Uruguayans, Chileans, ect.), and we certainly hope you do, there is a word you can use to describe yourself which will not offend and will probably make your new friend feel all warm and fuzzy inside. It's a Spanish word, so it won't help you too much with Canadians or Brazilians. But there are literally millions of people who would be simply charmed to hear you refer to yourself as estadounidense.
Unfortunately, it's not an easy word to pronounce. But it is useful.
I would like to add, to be perfectly clear, that we fully support use of the words America and American. We also support America and Americans whenever and however we can.