The phrase 'shall not exist in the US' is not a complete thought, so it is not a sentence.
You need a noun, such as:
Racism shall not exist in the US.
Communism shall not exist in the US.
The word sertainly does not exist.
we shall speak of this anon.
I suppose I shall have to answer this question, eventually.
The word knowned does not exist. The words known and reknown definitely do exist.
Some people believe that fairies exist.
I can reassure you that the word 'reasure' does not exist.
The word "mesovamemea" doesn't actually exist.
I shall deliver the box on Friday.
"I shall not dignify this insensitive remark with an answer"
I shall carry on, despite your ill will.
Depending on how it is used, exist can be a verb. If the sentence was, "The frog exists," it would be used as a verb. Let's say the sentence is "I do not like to exist." In this case, the word "exist" is used as the object of the preposition, and "like" is the verb.
The word "diagably" does not exist in the English language.