“America” refers to two continents, containing a wide assortment of nations, people, and languages. Write “US” or “USA.”
However, it is perfectly acceptable to call US residents “Americans.”
Don’t use “The States.” It’s far too casual, and it’s inaccurate: several parts of the US (eg, Guam, Puerto Rico, Washington, DC) do not have official statehood.
It’s counterintuitive, but “The United States of America,” and all of its short versions, is a singular noun. Ask a historian why.