States can sue but we know how long that process takes.I believe most of the abrogated responsibilities and authorities has been Congressional based.
States sue the US National government all the time when the US National Executive branch exceeds powers limited by the Tenth Amendment.
The States are no longer represented in the US National government with the passing of the 17th Amendment in 1913.
Seventeenth Amendment to the United States Constitution
The Seventeenth Amendment (Amendment XVII) to the United States Constitution established the popular election of United States senators by the people of the states. The amendment supersedes Article I, §3, Clauses 1 and 2 of the Constitution, under which senators were elected by state legislatures. It also alters the procedure for filling vacancies in the Senate, allowing for state legislatures to permit their governors to make temporary appointments until a special election can be held.
The amendment was proposed by the 62nd Congress in 1912 and became part of the Constitution on April 8, 1913 on ratification by three-fourths (36) of the state legislatures.