Someone who's slavish is either overly meek and obedient or copies someone else instead of thinking for themselves. If you write your story in the style of Ernest Hemingway, your English teacher may accuse you of slavish imitation.
In a dictatorship, rulers may expect citizens to be slavish, but in a democracy, no one should have to treat leaders in a slavish way. This meaning of slavish is the original 16th century one, from slave, "person who is the property of another person." The "servilely imitative, lacking originality or independence" meaning of slavish came later, around 1750.