I think it's worth pointing out that "the American continent" is not how geography is taught in the US. There's seven continents, one of which is North America and one of which is South America.
So you're making a point which only makes sense in Spanish, in the context of your own education. There's no ambiguity in the US since the only thing "America" can refer to is the USA.
I'm curious how it's taught for you about Europe and Asia. We learn those as separate continents, too, even though it's one land mass. For that matter, Africa is as connected to Asia as South America is to North America, but I'm almost certain you consider Africa its own continent, right?
“American” is the correct adjective in English to describe the United States’ people and government. There is simply no equivalent to the Spanish “estadounidense.”
Furthermore, North America and South America are considered to be separate continents, and if you want to refer to them both together, you say “the Americas”, plural.