One should refer to the US federal legislative body as "Congress"; capitalized and without 'the'.
States each have their own legislatures and can be referred to as "the state legislature of X" or "X state legislature". But they all have their own names, New York's is called the New York State Assembly. California simply calls theirs the California State Legislature. Either way works.
States each have their own legislatures and can be referred to as "the state legislature of X" or "X state legislature". But they all have their own names, New York's is called the New York State Assembly. California simply calls theirs the California State Legislature. Either way works.